Search

We show weighted non-autonomous L q (L p) maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let p , q ∈ (1 , ∞) and we consider coefficient functions in C t β + ε with values in C x α + ε subject to the parabolic relation 2 β + α = 1. If p < d α , we can likewise deal with spatial H x α + ε , d α regularity. The starting point for this result is a weak (p , q) -solution theory with uniform constants. Further key ingredients are a commutator argument that allows us to establish higher a priori spatial regularity, operator-valued pseudo differential operators in weighted spaces, and a representation formula due to Acquistapace and Terreni. Furthermore, we show p -bounds for semigroups and square roots generated by complex elliptic systems under a minimal regularity assumption for the coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we consider the principal spectral theory for the nonlocal dispersal eigenvalue problem (1) d ρ p [ ∫ Ω J ρ (x − y) u (y) d y − u (x) ] + κ ρ q [ ∫ Ω G ρ (x − y) u (y) d y − u (x) ] + a (x) u (x) = − λ u (x) , x ∈ Ω ¯ , where Ω ⊂ R N is a bounded smooth domain, J ρ (x) = ρ N J (ρ x) , G ρ (x) = ρ N G (ρ x) , the kernel functions J (x) and G (x) are nonnegative, J (x) is symmetric, the parameter ρ is positive, d , κ are positive, p , q are given constants, and a ∈ C (Ω ¯). We investigate the limiting behavior of the principal spectral point or generalized principal eigenvalue of (1) as ρ → ∞ and ρ → 0. When ρ ≫ 1 , the nonlocal dispersal operator u (⋅) ↦ d ρ p (∫ Ω J ρ (⋅ − y) u (y) d y − u (⋅)) + κ ρ q (∫ Ω G ρ (⋅ − y) u (y) d y − u (⋅)) + a (⋅) u (⋅) with ν 0 ≠ 0 behaves like the elliptic operator u (⋅) ↦ d c 0 ρ p − 2 Δ u − κ ρ q − 1 ν 0 ⋅ ∇ u + a (⋅) u (⋅) on Ω with Dirichlet boundary condition on ∂Ω, where c 0 = 1 2 N ∫ R N J (y) | y | 2 d y , ν 0 = 〈 ∫ R N G (x) x 1 d x , ∫ R N G (x) x 2 d x , ⋯ , ∫ R N G (x) x N d x 〉. The results obtained in the paper are novel when the dispersal kernel function G (x) is asymmetric. Moreover, when G is asymmetric with ν 0 ≠ 0 , and p ≥ 2 , q > p − 1 or p < 2 , q > 1 , it is seen that the limiting behavior of the generalized principal eigenvalue of (1) as ρ → ∞ is strikingly different from the limiting behavior of the principal eigenvalue of the elliptic operator u (⋅) ↦ d c 0 ρ p − 2 Δ u − κ ρ q − 1 ν 0 ⋅ ∇ u + a (⋅) u (⋅) on Ω with Dirichlet boundary condition on ∂Ω. The main results are used to study the asymptotic dynamics of nonlinear nonlocal dispersal problems. [ABSTRACT FROM AUTHOR]

Let L A = − div (A ∇) be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set Ω ⊆ R d. We prove that the maximal operator M A f = sup t > 0 ⁡ | exp ⁡ (− t L A) f | is bounded in L p (Ω) whenever A is p -elliptic in the sense of [11]. The relevance of this result is that, in general, the semigroup generated by − L A is neither contractive in L ∞ nor positive, therefore neither the Hopf–Dunford–Schwartz maximal ergodic theorem [16, Chap. VIII] nor Akcoglu's maximal ergodic theorem [2] can be used. We also show that if d ⩾ 3 and the domain of the sesquilinear form associated with L A embeds into L 2 ⁎ (Ω) with 2 ⁎ = 2 d / (d − 2) , then the range of L p -boundedness of M A improves to (r d / ((r − 1) d + 2) , r d / (d − 2)) , where r ⩾ 2 is such that A is r -elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator sup s , t > 0 ⁡ | T s A 1 T t A 2 f |. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ⁡ ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation Δ X 2 u = V u , where Δ X = Δ x + | x | 2 α Δ y (0 < α ≤ 1), with x ∈ R m , y ∈ R n , denotes the Baouendi-Grushin type subelliptic operators, and the potential V satisfies the strongly singular growth assumption | V | ≤ c 0 ρ 4 , where ρ = (| x | 2 (α + 1) + (α + 1) 2 | y | 2) 1 2 (α + 1) is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms. [ABSTRACT FROM AUTHOR]

A class of vector-valued elliptic operators with unbounded coefficients, coupled up to the second-order is investigated in the Lebesgue space L p (R d ; R m) with p ∈ (1 , ∞) , providing sufficient conditions for the generation of an analytic C 0 -semigroup T (t). Under further assumptions, a characterization of the domain of the infinitesimal generator is given. [ABSTRACT FROM AUTHOR]

We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to − L u = f (u) , where L is a linear uniformly elliptic operator and f is C 1 , such that the linearized equation − L − f ′ (u) has nonnegative principal eigenvalue. Our main result is an estimate for the L 2 + γ norm of the gradient of stable solutions vanishing on the flat part of a half-ball, for any nonnegative and nondecreasing f. This bound only requires the elliptic coefficients to be Lipschitz. As a consequence, our estimate continues to hold in general C 1 , 1 domains if we further assume the nonlinearity f to be convex. This result is new even for the Laplacian, for which a C 3 regularity assumption on the domain was needed. [ABSTRACT FROM AUTHOR]

In this paper, we study the estimates of resolvents R (λ , L ε) = (L ε − λ I) − 1 , where L ε = − div (A (x / ε) ∇) is a family of second elliptic operators with symmetric, periodic and oscillating coefficients defined on a bounded domain Ω with ε > 0. For 1 < p < ∞ , we will establish uniform L p → L p , L p → W 0 1 , p , W − 1 , p → L p and W − 1 , p → W 0 1 , p estimates by using the real variable method. Meanwhile, we use Green functions for operators L ε − λ I to study the asymptotic behavior of R (λ , L ε) and obtain convergence estimates in L p → L p , L p → W 0 1 , p norm. [ABSTRACT FROM AUTHOR]

We derive global W 2 , p estimates (with n ≤ p < ∞) for viscosity solutions to fully nonlinear elliptic equations under relaxed structural assumptions on the governing operator that are weaker than convexity and oblique boundary conditions as follows: { F (D 2 u , D u , u , x) = f (x) in Ω β (x) ⋅ D u (x) + γ (x) u (x) = g (x) on ∂ Ω , for f ∈ L p (Ω) and under appropriate assumptions on the data β , γ , g and Ω ⊂ R n. Our approach makes use of geometric tangential methods, which consist of importing "fine regularity estimates" from a limiting profile, i.e., the Recession operator, associated with the original second-order one via compactness and stability procedures. As a result, we pay special attention to the borderline scenario, i.e., f ∈ BMO p ⊋ L ∞. In such a setting, we prove that solutions enjoy BMO p type estimates for their second derivatives. Finally, as another application of our findings, we obtain Hessian estimates to obstacle-type problems under oblique boundary conditions and no convexity assumptions, which may have their own mathematical interest. A density result for a suitable class of viscosity solutions will also be addressed. [ABSTRACT FROM AUTHOR]

In this paper we study the Dirichlet problem in the ball for the so-called inframonogenic functions, i.e. the solutions of the sandwich equation ∂ x _ f ∂ x _ = 0 , where ∂ x _ stands for the Dirac operator in R m. The main steps in deriving our results are the establishment of some interior estimates for the first order derivatives of harmonic Hölder continuous functions and the proof of certain invariance property of the higher order Lipschitz class under the action of the Poisson integral. Using Mathematica we also implement an algorithm to find explicitly the solution of such a Dirichlet problem for a much wider class of partial differential equations in the ball of R 3 with polynomial boundary data. [ABSTRACT FROM AUTHOR]

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation d u d t = A u with strongly elliptic operator A on bounded and unbounded domains with white noise boundary data. Our main assumption is that the heat kernel of the corresponding homogeneous problem enjoys the Gaussian type estimates taking into account the distance to the boundary. Under mild assumptions about the domain, we show that A generates a C 0 -semigroup in weighted L p -spaces where the weight is an appropriate power of the distance to the boundary. We also prove some smoothing properties and exponential stability of the semigroup. Finally, we reformulate the Cauchy-Dirichlet problem with white noise boundary data as an evolution equation in the weighted space and prove the existence of Markovian solutions and invariant measures. [ABSTRACT FROM AUTHOR]

This article studies the unique continuation of the principal and inverse problem of a perturbed anisotropic fourth-order elliptic operator arising in elastic beam theory, where the principle part of the operator is modelled by the non-rotational symmetric operator ∑ j = 1 n D x j 4. We discuss the three-ball inequality, stability estimate, strong unique continuation principle, and finally the inverse boundary value problem of recovering the coefficients featuring the bending stiffness of the material. [ABSTRACT FROM AUTHOR]

In this article, we reproduce results of classical regularity theory of quasilinear elliptic equations in the divergence form, in the setting of Heisenberg Group. The considered cases encompass a very wide class of equations with isotropic growth conditions that are generalizations of the p -Laplacian and include equations with polynomial or exponential type growth. Some more general conditions have also been explored. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 , w be a Muckenhoupt A 2 (R n) weight, Ω a bounded Lipschitz domain of R n , and L : = − w − 1 div (A ∇ ⋅) the degenerate elliptic operator on Ω with the Dirichlet or the Neumann boundary condition. In this article, the authors establish the following weighted L p estimate for the Kato square root of L : ‖ L 1 / 2 (f) ‖ L p (Ω , v w) ∼ ‖ ∇ f ‖ L p (Ω , v w) for any f ∈ W 0 1 , p (Ω , v w) when L satisfies the Dirichlet boundary condition, or, for any f ∈ W 1 , p (Ω , v w) with ∫ Ω f (x) d x = 0 when L satisfies the Neumann boundary condition, where p is in an interval including 2, v belongs to both some Muckenhoupt weight class and the reverse Hölder class with respect to w , W 0 1 , p (Ω , v w) and W 1 , p (Ω , v w) denote the weighted Sobolev spaces on Ω, and the positive equivalence constants are independent of f. As a corollary, under some additional assumptions on w , via letting v : = w − 1 , the unweighted L 2 estimate for the Kato square root of L that ‖ L 1 / 2 (f) ‖ L 2 (Ω) ∼ ‖ ∇ f ‖ L 2 (Ω) for any f ∈ W 0 1 , 2 (Ω) when L satisfies the Dirichlet boundary condition, or, for any f ∈ W 1 , 2 (Ω) with ∫ Ω f (x) d x = 0 when L satisfies the Neumann boundary condition, are obtained. Moreover, as applications of these unweighted L 2 estimates, the unweighted L 2 regularity estimates for the weak solutions of the corresponding degenerate parabolic equations in Ω with the Dirichlet or the Neumann boundary condition are also established. [ABSTRACT FROM AUTHOR]

The paper studies a bounded symmetric operator A ε in L 2 (R d) with (A ε u) (x) = ε − d − 2 ∫ R d a ((x − y) / ε) μ (x / ε , y / ε) (u (x) − u (y)) d y ; here ε is a small positive parameter. It is assumed that a (x) is a non-negative L 1 (R d) function such that a (− x) = a (x) and the moments M k = ∫ R d | x | k a (x) d x , k = 1 , 2 , 3 , are finite. It is also assumed that μ (x , y) is Z d -periodic both in x and y function such that μ (x , y) = μ (y , x) and 0 < μ − ⩽ μ (x , y) ⩽ μ + < ∞. Our goal is to study the limit behaviour of the resolvent (A ε + I) − 1 , as ε → 0. We show that, as ε → 0 , the operator (A ε + I) − 1 converges in the operator norm in L 2 (R d) to the resolvent (A 0 + I) − 1 of the effective operator A 0 being a second order elliptic differential operator with constant coefficients of the form A 0 = − div g 0 ∇. We then obtain sharp in order estimates of the rate of convergence. [ABSTRACT FROM AUTHOR]

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form. [ABSTRACT FROM AUTHOR]

In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary, that is, a pair of the Neumann and Dirichlet data of the solution on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [13]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, the precise lower bound estimate of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem. [ABSTRACT FROM AUTHOR]

In this article we consider direct and inverse problems for α -stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only "one-sided operators" new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise. [ABSTRACT FROM AUTHOR]

We establish optimal boundary regularity results for viscosity solutions to second order fully nonlinear uniformly elliptic equations in the form F (D 2 u (x) , D u (x) , x) = f (x) in Ω. In particular, we obtain sharp estimates for borderline cases f ∈ L n (Ω) and f ∈ B M O (Ω). For source functions in B M O (Ω) , we obtain C 1 , L o g − L i p interior regularity for flat solutions of non-convex elliptic equations. As a consequence, we obtain C 1 , L o g − L i p estimates near the boundary; which again is an optimal estimate. [ABSTRACT FROM AUTHOR]

In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form − div (A (| ∇ u |) ∇ u) + B (| ∇ u |) = f (u) ; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin. [ABSTRACT FROM AUTHOR]

We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted p -Laplace term. A well chosen diffeomorphism between R and (− 1 , 1) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in L 1. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S +) -property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations with right-hand sides that have gradient dependence (so-called convection terms) under very general assumptions on the data. As a result of independent interest, we also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded. [ABSTRACT FROM AUTHOR]

This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nyström-based interpolation method. [ABSTRACT FROM AUTHOR]

We study elliptic and parabolic problems governed by the singular elliptic operators L = Δ x + D y y + c y D y − b y 2 in the half-space R + N + 1 = { (x , y) : x ∈ R N , y > 0 }. [ABSTRACT FROM AUTHOR]

In this paper we study the Neumann problem for a type of fully nonlinear second order elliptic partial differential equations on domains in C n without any curvature assumptions on the domain. [ABSTRACT FROM AUTHOR]

Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes k -vector-space Riesz potentials in R n with k < n , and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on k -dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in R n , or a solution operator for the transport equation in R n. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions. [ABSTRACT FROM AUTHOR]

Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms. When suitably scaled, graph Laplacians approach limiting continuum operators in the large data limit. Studying these limiting operators, therefore, sheds light on learning algorithms. This paper is devoted to the study of a parameterized family of divergence form elliptic operators that arise as the large data limit of graph Laplacians. The link between a three-parameter family of graph Laplacians and a three-parameter family of differential operators is explained. The spectral properties of these differential operators are analyzed in the situation where the data comprises of two nearly separated clusters, in a sense which is made precise. In particular, we investigate how the spectral gap depends on the three parameters entering the graph Laplacian, and on a parameter measuring the size of the perturbation from the perfectly clustered case. Numerical results are presented which exemplify the analysis and which extend it in the following ways: the computations study situations in which there are two nearly separated clusters, but which violate the assumptions used in our theory; situations in which more than two clusters are present, also going beyond our theory; and situations which demonstrate the relevance of our studies of differential operators for the understanding of finite data problems via the graph Laplacian. The findings provide insight into parameter choices made in learning algorithms which are based on weighted adjacency matrices; they also provide the basis for analysis of the consistency of various unsupervised and semi-supervised learning algorithms, in the large data limit. [ABSTRACT FROM AUTHOR]

In the present paper we prove estimates on subsolutions of the equation − A v + c (x) v = 0 , x ∈ D , where D ⊂ R d is a domain (i.e. an open and connected set) and A is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered "quantitative" versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with A and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on T 1 × S 3. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition (P) holds, in addition to an analytic Diophantine condition. [ABSTRACT FROM AUTHOR]

In the present paper, we study the following planar Choquard equation: { − Δ u + V (x) u = (I α ⁎ F (u)) f (u) , x ∈ R 2 , u ∈ H 1 (R 2) , where V (x) is an 1-periodic function, I α : R 2 → R is the Riesz potential and f (t) behaves like ± e β t 2 as t → ± ∞. A direct approach is developed in this paper to deal with the problems with both critical exponential growth and strongly indefinite features when 0 lies in a gap of the spectrum of the operator − △ + V. In particular, we find nontrivial solutions for the above equation with critical exponential growth, and establish the existence of ground states and geometrically distinct solutions for the equation when the nonlinearity has subcritical exponential growth. Our results complement and generalize the known ones in the literature concerning the positive potential V to the general sign-changing case, such as, the results of de Figueiredo-Miyagaki-Ruf (1995) [16] , of Alves-Cassani-Tarsi-Yang (2016) [4] , of Ackermann (2004) [1] , and of Alves-Germano (2018) [5]. [ABSTRACT FROM AUTHOR]

The triangular elliptic operators are natural extensions of the elliptic deformation of circular operators. We obtain a Brown measure formula for the sum of a triangular elliptic operator g α , β , γ with a random variable x 0 , which is ⁎-free from g α , β , γ with amalgamation over certain unital subalgebra. Let c t be a circular operator. We prove that the Brown measure of x 0 + g α , β , γ is the push-forward measure of the Brown measure of x 0 + c t by an explicitly defined map on C for some suitable t. We show that the Brown measure of x 0 + c t is absolutely continuous with respect to the Lebesgue measure on C and its density is bounded by 1 / (π t). This work generalizes earlier results on the addition with a circular operator, semicircular operator, or elliptic operator to a larger class of operators. We extend operator-valued subordination functions, due to Biane and Voiculescu, to certain unbounded operators. This allows us to extend our results to unbounded operators. [ABSTRACT FROM AUTHOR]

The quasilinear differential system x ′ = a x + b | y | p ⁎ − 2 y + k (t , x , y) , y ′ = c | x | p − 2 x + d y + l (t , x , y) is considered, where a , b , c and d are real constants with b 2 + c 2 > 0 , p and p ⁎ are positive numbers with (1 / p) + (1 / p ⁎) = 1 , and k and l are continuous for t ≥ t 0 and small x 2 + y 2. When p = 2 , this system is reduced to the linear perturbed system. It is shown that the behavior of solutions near the origin (0 , 0) is very similar to the behavior of solutions to the unperturbed system, that is, the system with k ≡ l ≡ 0 , near (0 , 0) , provided k and l are small in some sense. It is emphasized that this system can not be linearized at (0 , 0) when p ≠ 2 , because the Jacobian matrix can not be defined at (0 , 0). Our result will be applicable to study radial solutions of the quasilinear elliptic equation with the differential operator r − (γ − 1) (r α | u ′ | β − a u ′) ′ , which includes p -Laplacian and k -Hessian. [ABSTRACT FROM AUTHOR]

We study the well-posedness of Cauchy problems on the upper half space R + n + 1 associated to higher order systems ∂ t u = (− 1) m + 1 div m A ∇ m u with bounded measurable and uniformly elliptic coefficients. We address initial data lying in L p (1 < p < ∞) and BMO (p = ∞) spaces and work with weak solutions. Our main result is the identification of a new well-posedness class, given for p ∈ (1 , ∞ ] by distributions satisfying ∇ m u ∈ T m p , 2 , where T m p , 2 is a parabolic version of the tent space of Coifman–Meyer–Stein. In the range p ∈ [ 2 , ∞ ] , this holds without any further constraints on the operator and for p = ∞ it provides a Carleson measure characterization of BMO with non-autonomous operators. We also prove higher order L p well-posedness, previously only known for the case m = 1. The uniform L p boundedness of propagators of energy solutions plays an important role in the well-posedness theory and we discover that such bounds hold for p close to 2. This is a consequence of local weak solutions being locally Hölder continuous with values in spatial L l o c p for some p > 2 , what is also new for the case m > 1. [ABSTRACT FROM AUTHOR]

Maximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators L in R N and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author). [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in R N , and equipped with homogeneous Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

We study an eigenvalue problem involving a fully anisotropic elliptic differential operator in arbitrary Orlicz-Sobolev spaces. The relevant equations are associated with constrained minimization problems for integral functionals depending on the gradient of competing functions through general anisotropic N -functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are not even assumed to satisfy the so called Δ 2 -condition. The resulting analysis requires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces. [ABSTRACT FROM AUTHOR]

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type L u : = − r − θ (r α | u ′ (r) | β u ′ (r)) ′ , where θ , β ≥ 0 and α > 0 , are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the p -Laplacian and k -Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Pólya-Szegö principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality. [ABSTRACT FROM AUTHOR]

We give a direct analytic proof of the classical Boundary Harnack Principle for solutions to linear uniformly elliptic equations in either divergence or non-divergence form. [ABSTRACT FROM AUTHOR]

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators. [ABSTRACT FROM AUTHOR]

In this paper we derive a propagation of smallness result for a scalar second elliptic equation in divergence form whose leading order coefficients are Lipschitz continuous on two sides of a C 2 hypersurface that crosses the domain, but may have jumps across this hypersurface. Our propagation of smallness result is in the most general form regarding the locations of domains, which may intersect the interface of discontinuity. At the end, we also list some consequences of the propagation of smallness result, including stability results for the associated Cauchy problem, a propagation of smallness result from sets of positive measure, and a quantitative Runge approximation property. [ABSTRACT FROM AUTHOR]

We consider elliptic second order partial differential operators with Lipschitz continuous leading order coefficients on finite cubes and the whole Euclidean space. We prove quantitative sampling and equidistribution theorems for eigenfunctions. The estimates are scale-free, in the sense that for a sequence of growing cubes we obtain uniform estimates. These results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum, and an uncertainty relation (aka spectral inequality) for short energy interval spectral projectors. Several applications including random operators are discussed. In the proof we have to overcome several challenges posed by the variable coefficients of the leading term. [ABSTRACT FROM AUTHOR]

In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension 2. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree 2. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin. [ABSTRACT FROM AUTHOR]

This paper concerns the Liouville type result for the general semilinear elliptic equation (S) a i j (x) ∂ i j u (x) + K q i (x) ∂ i u (x) + f (x , u (x)) = 0 a.e. in R N , where f is of the KPP-monostable nonlinearity, as a continuation of the previous works of the second author [31,32]. The novelty of this work is that we allow f s (x , 0) to be sign-changing and to decay fast up to a Hardy potential near infinity. First, we introduce a weighted generalized principal eigenvalue and use it to characterize the Liouville type result for Eq. (S) that was proposed by H. Berestycki. Secondly, if (a i j) is the identity matrix and q is a compactly supported divergence-free vector field, we find a condition that Eq. (S) admits no positive solution for K > K ⋆ , where K ⋆ is a certain positive threshold. To achieve this, we derive some new techniques, thanks to the recent results on the principal spectral theory for elliptic operators [14] , to overcome some fundamental difficulties arising from the lack of compactness in the domain. This extends a nice result of Berestycki-Hamel-Nadirashvili [5] on the limit of eigenvalues with large drift to the case without periodic condition. Lastly, the well-posedness and long time behavior of the evolution equation corresponding to (S) are further investigated. The main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for (S) holds or not. [ABSTRACT FROM AUTHOR]

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [12] and [20]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p -evolution equations for higher order operators on R n or on groups, the results already being new in all these cases. [ABSTRACT FROM AUTHOR]

It has been recently shown that complex two-dimensional (2D) potentials V ɛ (x , y) = V (y + i ɛ η (x)) can be used to emulate non-Hermitian matrix gauge fields in optical waveguides. Here x and y are the transverse coordinates, V (y) and η (x) are real functions, ɛ > 0 is a small parameter, and i is the imaginary unit. The real potential V (y) is required to have at least two discrete eigenvalues in the corresponding 1D Schrödinger operator. When both transverse directions are taken into account, these eigenvalues become thresholds embedded in the continuous spectrum of the 2D operator. Small nonzero ɛ corresponds to a non-Hermitian perturbation which can result in a bifurcation of each threshold into an eigenvalue. Accurate analysis of these eigenvalues is important for understanding the behavior and stability of optical waves propagating in the artificial non-Hermitian gauge potential. Bifurcations of complex eigenvalues out of the continuum is the main object of the present study. Using recent mathematical results from the rigorous analysis of elliptic operators, we obtain simple asymptotic expansions in ɛ that describe the behavior of bifurcating eigenvalues. The lowest threshold can bifurcate into a single eigenvalue, while every other threshold can bifurcate into a pair of complex eigenvalues. These bifurcations can be controlled by the Fourier transform of function η (x) evaluated at certain isolated points of the reciprocal space. When the bifurcation does not occur, the continuous spectrum of 2D operator contains a quasi-bound-state which is characterized by a strongly localized central peak coupled to small-amplitude but nondecaying tails. The analysis is applied to the case examples of parabolic and double-well potentials V (y). In the latter case, the bifurcation of complex eigenvalues can be dampened if the two wells are widely separated. • A class of 2D complex optical potentials to emulate non-Hermitian gauge fields. • Asymptotics for complex eigenvalues bifurcating from the continuum in 2D waveguides. • Complex eigenvalues limit the applicability of the gauge field theory. • Control and suppression of unstable eigenvalues by tuning the gauge field shape. • A mechanism for systematic generation of non-Hermitian BIC states. [ABSTRACT FROM AUTHOR]

Mathematical analysis on electromagnetic waves in photonic graphene, a photonic topological material which has a honeycomb structure, is one of the most important current research topics. By modulating the honeycomb structure, numerous topological phenomena have been observed recently. The electromagnetic waves in such media are generally described by the 2-dimensional wave equation. It has been shown that the corresponding elliptic operator with a honeycomb material weight has Dirac points in its dispersion surfaces. In this paper, we study the time evolution of the wave packets spectrally concentrated at such Dirac points in a modulated honeycomb material weight. We prove that such wave packet dynamics is governed by the Dirac equation with a varying mass in a large but finite time. Our analysis provides mathematical insights to those topological phenomena in photonic graphene. [ABSTRACT FROM AUTHOR]

In this paper, we obtain sharp restricted weak-type uniform Sobolev inequalities for a class of elliptic operators P (D). In particular, we improve a recent result of Sikora, Yan and Yao [24] to the endpoint case. The sharpness is proved by constructing explicit counterexamples. As applications, we establish Stein-Tomas type inequalities for H = P (D) + V with certain class of potentials V. [ABSTRACT FROM AUTHOR]

We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable Hölder coefficients when the underlying domain is bounded and has a C 1 + κ -boundary for some κ > 0. We also prove a number of other results such as gradient estimates for heat kernels and Green functions G of elliptic operators with possibly complex-valued coefficients. We establish Hölder continuity of ∇ x ∇ y G up to the boundary. These results are used to prove L p -estimates for commutators of Dirichlet-to-Neumann operators on the boundary of C 1 + κ -domains. Such estimates are the keystone in our approach for the Poisson bounds. [ABSTRACT FROM AUTHOR]

Let Ω ε be a bounded domain in R N with small holes. In this paper, we study the existence of solutions for the following problem { (− Δ) s u = u 2 s ⁎ − 1 , u > 0 , in Ω ε , u = 0 , in R N ∖ Ω ε , where 0 < s < 1 and 2 s ⁎ = 2 N N − 2 s. We construct solutions which blow up like a volcano near the center of each hole in Ω ε [ABSTRACT FROM AUTHOR]