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]

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L 2 - and H 1 -norms are derived for the problem with initial data u 0 ∈ H 0 1 (Ω) ∩ H 2 (Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H 1 -norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L ∞ error estimate is obtained for 2D problems that too with the initial condition is in H 0 1 (Ω) ∩ H 2 (Ω). All results proved in this paper are valid uniformly as α → 1 − , where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

This article sets forth results on the existence, non-existence, uniqueness, and regularities properties, as well as boundary behavior of solutions for singular systems involving mixed local and non-local elliptic operators (see System (S) below). More precisely, we first establish a new weak comparison principle for a singular equation. Afterward, we discuss the non-existence of positive classical solutions, as well as construct suitable ordered pairs of sub-solutions and super-solutions. This allows us to obtain the existence of a pair of positive weak solutions for System (S) by employing Schauder's fixed-point theorem in the associated conical shell. Finally, we adapt a method of Krasnoselsky to establish the uniqueness of such a positive pair of solutions. [ABSTRACT FROM AUTHOR]

We give a characterization of the singular points of the free boundary ∂{u>0} for viscosity solutions of the nonlinear equation F(D²u)=-χ {u>0}, where F is a fully nonlinear elliptic operator and χ is the characteristic function. This equation models the propagation of a nerve impulse along an axon. We analyze the structure of the free boundary ∂{u>0} near the singular points where u and ∇u vanish simultaneously. Our method uses the stratification approach developed in Dipierro and the author's 2018 paper. In particular, when n=2 we show that near a flat singular free boundary point, ∂{u>0} is a union of four C 1 arcs tangential to a pair of crossing lines. [ABSTRACT FROM AUTHOR]

We prove new multiplicity results for some nonlocal critical growth elliptic problems in bounded domains. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter λ>0$\lambda > 0$. In particular, the number of solutions goes to infinity as λ→∞$\lambda \rightarrow \infty$. We also give an explicit lower bound on λ$\lambda$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues of the associated nonlocal elliptic operator. The proofs are based on an abstract critical point theorem. [ABSTRACT FROM AUTHOR]

This paper studies the superlinear convergence of Krylov iterations under the streamline‐diffusion preconditioning operator for convection‐dominated elliptic problems. First, convergence results are given involving the diffusion parameter ε$$ \varepsilon $$. Then the limiting case ε=0$$ \varepsilon =0 $$ is studied on the operator level, and the convergence results are extended to this situation under some conditions, in spite of the lack of compactness of the perturbation operators. An explicit rate is also given. [ABSTRACT FROM AUTHOR]

The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p-Laplacian type when the size of the domain goes to infinity in different directions. [ABSTRACT FROM AUTHOR]

The fractional differential equation L^\beta u = f posed on a compact metric graph is considered, where \beta >0 and L = \kappa ^2 - \nabla (a\nabla) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients \kappa,a. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L^{-\beta }. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L_2(\Gamma \times \Gamma)-error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for {L = \kappa ^2 - \Delta, \kappa >0} are performed to illustrate the results. [ABSTRACT FROM AUTHOR]

We propose a notion of stability for constant k$k$‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem. [ABSTRACT FROM AUTHOR]

We prove Schauder estimates for elliptic and parabolic problems governed by the degenerate operator ℒ = Δ x + D y ⁢ y + c y ⁢ D y , in the half-space Ω = { (x , y) : x ∈ ℝ N , y > 0 } , under Neumann boundary conditions at y = 0 . [ABSTRACT FROM AUTHOR]

This paper is devoted to dealing with a kind of new Kirchhoff-type problem in RN that involves a general double-phase variable exponent elliptic operator ϕ. Specifically, the operator ϕ has behaviors like |τ|q(x)−2 τ if |τ| is small and like |τ| p(x)−2 τ if |τ| is large, where 1 < p(x) < q(x) < N. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature. [ABSTRACT FROM AUTHOR]

In this article, we study the numerical corroboration of a variational model governed by a fourth-order elliptic operator that describes the deformation of a linearly elastic flexural shell subjected not to cross a prescribed flat obstacle. The problem under consideration is modelled by means of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Sobolev space and is known to admit a unique solution. Qualitative and quantitative numerical experiments corroborating the validity of the model and its asymptotic similarity with Koiter's model are also presented. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

We consider a second-order elliptic operator with sufficiently smooth variable coefficients in an arbitrary two-dimensional domain with fast oscillating boundary under the assumption that the oscillation amplitude is small. The structure of the oscillations is fairly arbitrary and no periodicity or local periodicity conditions are imposed. The oscillating boundary is divided into two components with the Dirichlet boundary condition posed on one of the components and the Neumann condition on the other. Such mixed boundary conditions are preserved under homogenization; as a result, the functions in the domain of the homogenized operator have weak power-law singularities. Despite these singularities, we succeed to modify the technique from our previous papers appropriately and to prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and estimate the convergence rate. [ABSTRACT FROM AUTHOR]

Given an action of a discrete group on a closed smooth manifold, we consider the class of nonlocal elliptic operators generated by pseudodifferential operators on the manifold and shift operators. The operators in this class are Fredholm under suitable ellipticity conditions. In the present paper, we give an index formula for these operators for the case of the group , where is an arbitrary finite group. To solve this index problem, we explicitly write the contribution to the index by the fixed points of finite-order elements of the group. The index formula is stated in terms of characteristic classes in periodic cyclic cohomology. [ABSTRACT FROM AUTHOR]

In this work, we investigate a divergence-free, nonconforming virtual element method for the numerical approximation of the Stokes eigenvalue problems using polygonal unstructured meshes on polygonal domain. By employing an elliptic projection operator, we "enhance" the definition of the virtual element space to make the L 2 -projection operator computable with optimal order. Furthermore, we prove the optimal order of convergence of the eigenfunction approximation and the double order of convergence of the eigenvalue approximation through compactness arguments for the solution operator (Babuška-Osborn Theory). Finally, we present a set of numerical experiments to assess the good approximation behavior of the method and show its computational performance. [ABSTRACT FROM AUTHOR]

In this research, we study certain characteristics of the solution for an equation involving uniformly elliptic nonlocal operator by establishing various maximum principles in bounded and unbounded regions. Additionally, we don't need the conditions that the narrow region principle's bound assumption and decay in the unbounded domain. In order to get the monotonicity of the solution, we use several maximum principles and averaging effects to overcome the challenges that generated by the uniformly elliptic nonlocal operator and fractional‐order variable t$$ t $$. Of course, while investigating the qualitative characteristics of the equation, we also provide some estimates of auxiliary function. For the study of additional nonlocal operators, we believe the method of uniformly elliptic nonlocal operators will also be helpful. [ABSTRACT FROM AUTHOR]

We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨ · ⟩ and the corresponding linear elliptic operator - ∇ · a ∇ . In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨ · ⟩ . We make this point by investigating the bias (or systematic error), i.e., the difference between a hom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O (L - 1) . In case of a suitable periodization of ⟨ · ⟩ , we rigorously show that it is O (L - d) . In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨ · ⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the small data global existence of solutions to the semilinear damped wave equation$ \partial_t^2 u + Au + \partial_t u = |u|^{p-1}u $on a measure space $ X $ with a self-adjoint operator $ A $ on $ L^2(X) $. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian. [ABSTRACT FROM AUTHOR]

In 1954, Hirzebruch reported a problem posed by Kodaira and Spencer: on compact almost complex manifolds, is the dimension h∂̄p,q of the kernel of the Dolbeault Laplacian independent of the choice of almost Hermitian metric? In this paper, we review recent progresses on the original problem and we introduce a similar one: on compact almost complex manifolds, find a generalization of Bott–Chern and Aeppli numbers which is metric-independent. We find a solution to our problem valid on almost Kähler 4-manifolds. [ABSTRACT FROM AUTHOR]

Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context, in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs. [ABSTRACT FROM AUTHOR]

We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyze Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients, e.g., integration by quasi--Monte Carlo methods. [ABSTRACT FROM AUTHOR]

In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace--Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of high-order curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order r > 1 and to the finite element degree k > 1, whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter k = r. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters k and r. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case. [ABSTRACT FROM AUTHOR]

In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p-Laplacian-type diffusion operator with 1 < p < ∞ . This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi's truncation and freezing coefficient methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the $ (\rho,\delta) $ (ρ , δ) -Hörmander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case $ \rho =\delta. $ ρ = δ. Interpolation properties between Besov spaces and Triebel–Lizorkin spaces are also investigated. [ABSTRACT FROM AUTHOR]

We provide a semi-constructive criterion for ellipticity of the differential operator on the Heisenberg group H 1. [ABSTRACT FROM AUTHOR]

In this paper we consider the vector-valued Schrödinger operator − Δ + V , where the potential term V is a matrix-valued function whose entries belong to L loc 1 (R d) and, for every x ∈ R d , V (x) is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in L 1 (R d , R m). Assuming further that the minimal eigenvalue of V belongs to some reverse Hölder class of order q ∈ (1 , ∞) ∪ { ∞ } , we obtain maximal inequality in L p (R d , R m) , for p in between 1 and some q , and generation results. [ABSTRACT FROM AUTHOR]

A theorem of Llarull says that if a smooth metric g$g$ on the n$n$‐sphere Sn$\mathbb {S}^n$ is bounded below by the standard round metric and the scalar curvature Rg$R_g$ of g$g$ is bounded below by n(n−1)$n (n - 1)$, then the metric g$g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound Rg⩾n(n−1)$R_g \geqslant n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature Rg$R_g$. We utilize two methods: spinor and spacetime harmonic function. [ABSTRACT FROM AUTHOR]

In this paper we study critical sets of solutions uε$u_\varepsilon$ of second‐order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first‐order correctors, we show that the (d−2)$(d-2)$‐dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε$\varepsilon$, provided that doubling indices for solutions are bounded. The key step is an estimate of "turning" of an approximate tangent map, the projection of a non‐constant solution uε$u_\varepsilon$ onto the subspace of spherical harmonics of order ℓ$\ell$, when the doubling index for uε$u_\varepsilon$ on a sphere ∂B(0,r)$\partial B(0, r)$ is trapped between ℓ−δ$\ell -\delta$ and ℓ+δ$\ell +\delta$, for r$r$ between 1 and a minimal radius r∗≥C0ε$r^*\ge C_0\varepsilon$. This estimate is proved by using harmonic approximation successively. With a suitable L2$L^2$ renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the following nonlinear elliptic problem A (u) = ρ (u) | ∇ φ | 2 in Ω , div (ρ (u) ∇ φ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω. where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L1(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique. [ABSTRACT FROM AUTHOR]

We establish Lyapunov‐type inequality for equations concerning general class of second‐order non‐symmetric elliptic operators with singular coefficients. Our approach is based on the probabilistic representation of solutions and stochastic calculus. We also discuss a Lyapunov‐type inequality for equations pertaining to second‐order symmetric operator with some regularity assumptions on the coefficients and a nonlinear Neumann boundary condition. [ABSTRACT FROM AUTHOR]

This research studies the inverse boundary value problem for fractional elliptic equation of Tricomi–Gellerstedt–Keldysh type and obtains a condition stability result. To recover the continuous dependence of the solution on the measurement data, a generalized Tikhonov regularization method based on ill-posedness analysis is constructed. Under the a priori and a posterior selection rules for the regularization parameter, corresponding Hölder type convergence results are obtained. On this basis, this thesis verifies the simulation effect of the generalized Tikhonov method through numerical examples. The examples show that the method performs well in dealing with the problem under consideration. [ABSTRACT FROM AUTHOR]

We study spectral stability estimates for the Dirichlet eigenvalues of the Laplace operator in bounded two-connected planar domains. We propose a method based on the conformal analysis of elliptic operators. Owing to this method, it is possible to obtain spectral stability estimates in domains with nonrectifiable boundaries. [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]

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like | E (x) | | ∇ u | , where E (x) is a vector field belonging to a suitable Morrey-type space. Our operator arises as a stationary equation of diffusion–advection problems. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem. To obtain our main results, we use the weak maximum principle and the same a priori estimates. [ABSTRACT FROM AUTHOR]

We consider the quasilinear diffusion problem of u=(u1,...,um)$\bm {u}=(u^1,\ldots ,u^m)$u′+Π(t,x,u,Σu)Au=f(t,x,u,Σu)in]0,T[×Ω,u=0in]0,T[×Ωc,u(0,·)=u0(·)inΩ$$\begin{equation*} \hspace*{52pt}\begin{cases} \bm{u}'+\Pi(t,x,\bm{u},\Sigma \bm{u})\mathbb{A}\bm{u}=\bm{f}(t,x,\bm{u},\Sigma \bm{u})&\text{ in }]0,T[\times\Omega, \\ \bm{u}=\bm{0}&\text{ in }]0,T[\times\Omega^c, \\ \bm{u}(0,\cdot)=\bm{u}_0(\cdot)&\text{ in }\Omega \end{cases} \end{equation*}$$for an open set Ω⊂Rn$\Omega \subset \mathbb {R}^n$, u0∈H0s(Ω):=[H0s(Ω)]m$\bm {u}_0\in \mathbf {H}^s_0(\Omega):=[H^s_0(\Omega)]^m$, for 0

We consider a class of noncooperative Schrödinger–Kirchhof–type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration–compactness principle for weighted‐variable exponent Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz. [ABSTRACT FROM AUTHOR]

In this study, we investigate the m-th order elliptic operators on n-dimensional bounded domain Ω ⊂ Rn with discontinuous coefficients in the rearrangement-invariant Sobolev space WXm (Ω). In general, the considered rearrangement-invariant spaces are not separable, so the use of classical methods in these spaces requires substantial modification of classical methods and a lot of preparation, concerning correctness of substitution operator, problems related to the extension operator in such spaces, etc. For this purpose, the corresponding separable subspaces of these spaces, in which the set of compact supported infinitely differentiable functions is dense, are introduced based on the shift operator. We establish interior Schauder-type estimates in the above subspaces. Note that Lebesgue spaces Lp (Ω), grand-Lebesgue spaces, Marcinkiewicz spaces, weak-type Lpw spaces, etc. are also covered by such spaces. [ABSTRACT FROM AUTHOR]

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator L directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function ζ (s) = Tr L − s and the heat trace Θ (t) = Tr exp (− t L). The kernel U (t ; x , x ′) of the heat semigroup exp (− t L) , called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as Tr f (t L) , that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as Tr exp (− t L +) exp (− s L −) , that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods. [ABSTRACT FROM AUTHOR]

We consider an inverse scattering problems for the biharmonic Schrödinger operator Δ2 + A · ∇ + V in three dimensions. By the Helmholtz decomposition, we take A = ∇p + ∇ ×ψ. The main contributions of this work are twofold. First, we derive a stability estimate of determining the divergence-free part ∇ ×ψ of A by far-field data at multiple wavenumbers. As a consequence, we further derive a quantitative stability estimate of determining − 1 2 ∇ ⋅ A + V. Both the stability estimates improve as the upper bound of the wavenumber increases, which exhibit the phenomenon of increased stability. Second, we obtain the uniqueness of recovering both A and V by partial far-field data. The analysis employs scattering theory to obtain an analytic domain and an upper bound for the resolvent of the fourth order elliptic operator. Notice that due to an obstruction to uniqueness, the corresponding results do not hold in general for the Laplacian, i.e., Δ + A · ∇ + V. This can be explained by the fact that the resolvent of the biharmonic operator enjoys a faster decay estimate with respect to the wavenumber compared with the Laplacian. [ABSTRACT FROM AUTHOR]

By using a variant of the variational perturbation techniques introduced by Rabinowitz [Multiple critical points of perturbed symmetric functionals. Tran Amer Math Soc. 1982;272:753–769], we build a new non-smooth symmetric Mountain Pass Theorem without symmetric condition. This new abstract result can be applied to generalized quasi-linear elliptic equations with small perturbations. Our results extend the results concerning a semi-linear elliptic equation made by Li-Liu [Perturbations from symmetric elliptic boundary value problems. J Differ Equ. 2002;185:271–280]. [ABSTRACT FROM AUTHOR]

A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N , where 푁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials. [ABSTRACT FROM AUTHOR]

In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting of the sum of two terms, one of which is linear and the other is nonlinear. The well-posedness of the direct problem is examined and the results are used to investigate the stability of an inverse problem of determining a function in the linear part of the source. The main tools in our study are the generalized eigenfunction expansions theory for nonlinear fractional diffusion equations, contraction mapping, Young's convolution and generalized Grönwall's inequalities. We present a stability estimate for the solution of the inverse source problem by means of observation data at a given point in the domain. [ABSTRACT FROM AUTHOR]

In this paper, we explore the ball convergence properties of enhanced Phragmén–Lindelöf type methods for solving the Dirichlet problem with an elliptic operator. By placing requirements on the elliptic operator and auxiliary points, we characterize the ball limit of a series of elliptic operators in non‐divergence form. Considering all dimensions, it becomes apparent that dealing solely with measurable and bounded coefficients for this class of operators is insufficient, necessitating additional regularity assumptions on them. We find the radii of convergence balls using recurrence relations, so as to guarantee the convergence of iterative reconstruction techniques, beginning from any point inside the ball centered on an auxiliary point. [ABSTRACT FROM AUTHOR]

We derive an expression for the spectral determinant of a second-order elliptic differential operator T defined on the whole real line, in terms of the Wronskians of two particular solutions of the equation T u = 0 . Examples of application of the resulting formula include the explicit calculation of the determinant of harmonic and anharmonic oscillators with an added bounded potential with compact support. [ABSTRACT FROM AUTHOR]

This is the third paper in a series of three dealing with sums of squares and hypoellipticity in the infinitely degenerate regime.We establish a C2,d generalization of M. Christ's smooth sum of squares theorem, and then use a bootstrap argument with the sum of squares decomposition for matrix functions, obtained in our second paper of this series, to prove a hypoellipticity theorem that generalizes some cases of the results of Christ, Hoshiro, Koike, Kusuoka and Stroock and Morimoto for sums of squares, and of Fedıi and Kohn for degeneracies not necessarily a sum of squares. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of weak solutions for a class of degenerate elliptic Dirichlet problems with critical nonlinearity and a logarithmic perturbation, i.e. 0.2 { - (Δ x u + (α + 1) 2 | x | 2 α Δ y u) = u Q + 2 Q - 2 + λ u log u 2 , u = 0 on ∂ Ω , where (x , y) ∈ Ω ⊂ R N = R m × R n with m ≥ 1 , n ≥ 0 , Ω ∩ { x = 0 } ≠ ∅ is a bounded domain, the parameter α ≥ 0 and Q = m + n (α + 1) denotes the "homogeneous dimension" of R N . When λ = 0 , we know that from [23] the problem (0.2) has a Pohožaev-type non-existence result. Then for λ ∈ R \ { 0 } , we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions. [ABSTRACT FROM AUTHOR]

In the unit disc, we consider the Dirichlet problem for second order elliptic system with only higher order constant coefficients. We establish the unique solvability of this problem under the assumption that the problem is Fredholm, and obtain an explicit formula for the solution. [ABSTRACT FROM AUTHOR]

Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue—much in the spirit of earlier results by Donsker–Varadhan and Bañuelos–Carrol—as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including p$p$‐Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments. [ABSTRACT FROM AUTHOR]

This paper concerns elliptic problems involving supercritical Sobolev growth without the (AR) condition and with a mixed boundary Dirichlet-Neumann type condition. The conditions imposed on the nonlinearity considered here generalizes several previous papers, including that presented in the work that inspired this paper, due to Peral and Colorado, in 2003. Beyond that, we present some complementary results, concerning the non-existence of solutions to a class of elliptic problems and a comparison result inspired by the case of Dirichlet boundary conditions, presented by the work of Ambrosetti, Brezis and Cerami. [ABSTRACT FROM AUTHOR]

In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys O (1 k 2 ) iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm. [ABSTRACT FROM AUTHOR]