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Studying PDEs on the sets of complicated structure and, in particular, stratified sets has been gaining popularity since recently. This article establishes some local Aleksandrov–Bakelman–Krylov type maximum estimates for solutions to linear elliptic and parabolic second-order equations on a book-type stratified set. [ABSTRACT FROM AUTHOR]

We obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we give eigenvalue estimates in the Gaussian shrinking soliton, and we find a domain that makes the behavior of these estimates similar to the estimates for the case of the Laplacian. Moreover, we also give an answer to the generalized conjecture of Pólya. [ABSTRACT FROM AUTHOR]

We consider the homogeneous time-fractional diffusion equation @?.u - u0/.t / C Au.t/D 0; t -a.e. Here, @? denotes the Riemann-Liouville fractional derivative of order = 2 .0; 1/with respect to time and A is a uniformly elliptic operator on Rd. We prove decay estimates in time and other regularity properties for the solution of the above equation. [ABSTRACT FROM AUTHOR]

In this work, it is considered an elliptic operator L of mth order with nonsmooth coefficients in a non-standard grand Sobolev space W q) m (Ω) on a bounded domain Ω ⊂ R n generated by the norm of the grand Lebesgue space L q) (Ω). Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in W q) m (Ω) and also establish interior Schauder-type estimates for these spaces. These estimates play the main role in establishing the Fredholmness of the Dirichlet problem for the equation Lu = f. The considered spaces are not separable, infinitely differentiable functions are not dense in them, and therefore many classical methods concerning Sobolev spaces are not applicable in this case. Nevertheless, it is possible to obtain the corresponding results under the assumption that the coefficients of the principal terms of the operator L are continuous, and the rest are essentially bounded in Ω. [ABSTRACT FROM AUTHOR]

The steelpan is a pitched percussion instrument that takes the form of a concave bowl with several localized dimpled regions of varying curvature. Each of these localized zones, called notes, can vibrate independently when struck, and produce a sustained tone of a well-defined pitch. While the association of the localized zones with individual notes has long been known and exploited, the relationship between the shell geometry and the strength of the mode confinement remains unclear. Here, we explore the spectral properties of the steelpan modelled as a vibrating elastic shell. To characterize the resulting eigenvalue problem, we generalize a recently developed theory of localization landscapes for scalar elliptic operators to the vector-valued case, and predict the location of confined eigenmodes by solving a Poisson problem. A finite-element discretization of the shell shows that the localization strength is determined by the difference in curvature between the note and the surrounding bowl. In addition to providing an explanation for how a steelpan operates as a two-dimensional xylophone, our study provides a geometric principle for designing localized modes in elastic shells. [ABSTRACT FROM AUTHOR]

Suppose that the set of square-integrable solutions of an elliptic equation which have a value at some given point equal to c is not empty. Then there is exactly one element with minimal L 2 -norm. Moreover, it is shown that this minimal element depends continuously on a domain of integration, i.e., on the set on which our solutions are defined, and on a weight of integration, i.e., on the deformation of an inner product. The theorems are proved using the theory of reproducing kernels and Hilbert spaces of square-integrable solutions of elliptic equations. We prove the existence of such a reproducing kernel using theory of Sobolev spaces. We generalize the well-known Ramadanov theorem. This is done in three different ways. Two of them are similar to the techniques used by I. Ramadanov and M. Skwarczyński(see [11, 14, 13]) , while the third method using weak convergence is new. Moreover, we show that our reproducing kernel depends continuously on a weight of integration. The idea of using the minimal norm property in such a proof is novel and, which is important, it needs the convergence of weights only almost everywhere. [ABSTRACT FROM AUTHOR]

-invariants for a class of parameter-dependent nonlocal operators associated with an isometric action of a discrete group of polynomial growth on a smooth closed manifold are studied. The -invariant is defined as the regularization of the winding number. The formula for the variation of the -invariant when the operator changes is obtained. The results are based on the study of asymptotic expansions of traces of parameter-dependent nonlocal operators. [ABSTRACT FROM AUTHOR]

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type. [ABSTRACT FROM AUTHOR]

A higher order elliptic operator with non-smooth coeficients in Morrey-Sobolev spaces on a bounded domain in Rn is considered. These spaces are nonseparable, and infinite differentiable functions are not dense in them. For this reason, the classical methods of establishing interior (and other) estimates with respect to these operators, the possibility of extending functions (with a bounded norm), determining the trace and results associated with this concept, etc. are not applicable in Morrey-Sobolev spaces, and to establish these facts a different research approach should be chosen. This paper focuses on these issues. An extension theorem is proved, the trace of a function in a Morrey-Sobolev space on a (n = 1) dimensional smooth surface is defined, a theorem on the existence of a strong solution in the small is proved, an interior Schauder-type estimate in Morrey-Sobolev spaces is established. A constructive characterization of the space of traces of functions from the Morrey-Sobolev space is given, which differs from the characterization given earlier by S. Campanato. [42] who offered a different characterization of the space of traces based on different considerations. It should be noted that the approach proposed in this work differs from the classical approach to determining the trace. [ABSTRACT FROM AUTHOR]

Let be a smooth domain in (not necessarily bounded), and let be a linear elliptic differential operator of order with singular coefficients acting in . Under some assumptions of singularity for the coefficients of , we consider the Friedrichs extension and study the convergence of spectral expansions in Sobolev spaces. [ABSTRACT FROM AUTHOR]

Stochastic algorithms for solving the Dirichlet boundary value problem for a second-order elliptic equation with coefficients having a discontinuity on a smooth surface are considered. It is assumed that the solution is continuous and its normal derivatives on the opposite sides of the discontinuity surface are consistent. A mean value formula in a ball (or an ellipsoid) is proposed and proved. This formula defines a random walk in the domain and provides statistical estimators (on its trajectories) for finding a Monte Carlo solution of the boundary value problem at the initial point of the walk. [ABSTRACT FROM AUTHOR]

The article is devoted to the study of the global geometry of symmetric and skew-symmetric higher order tensors on complete Riemannian manifolds using second-order elliptic operators, which are constructed on the basis of Stein-Weiss operators. [ABSTRACT FROM AUTHOR]

The existence, uniqueness, and regularity properties and Strichartz-type estimates for the solution of an integral-type initial value problem for linear and nonlinear Schrödinger equations with the general elliptic leading part are obtained. [ABSTRACT FROM AUTHOR]

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the Sp (4 , ℝ) -Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the 2 g − 3 exceptional components of the maximal Sp (4 , ℝ) -Higgs bundle moduli space, which correspond to components solely consisting of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by Guichard and Wienhard. [ABSTRACT FROM AUTHOR]

Let U ⊆ Rn (res. D ⊆ Rn) be an open (res. a compact) subset, and let L be an elliptic operator defined on C²(U,R) (res. C²(D,R)). In the present paper, we are going to extend the maximum principle for the function f ∈ C²(U,R) (res. f ∈ C²(D,R)) satisfying the equation Lf = ε, where ε is a real everywhere nonzero continuous function on U (res. D). Finally, we obtain some applications in mathematics and physics. [ABSTRACT FROM AUTHOR]

For an arbitrary-rank vector bundle over a projective manifold, J.-P. Demailly proposed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor to settle a conjecture of Griffiths on the equivalence of Hartshorne ampleness and Griffiths positivity. In this article, we have studied two proposed systems and proved that these equations have smooth solutions for the Vortex bundle using the continuity method. [ABSTRACT FROM AUTHOR]

In this paper we develop and investigate numerical algorithms for solving equations of the first order for the fractional powers of discrete elliptic operators: A α y = φ , 0 < α < 1 , for φ ∈ V h with V h a finite element or discrete approximation space. The main goal is to consider two different approaches to construct efficient high order time stepping schemes for the implementation of the Cauchy problem method (Vabishchevich, 2015). The first approach is based on a solution of the related time-dependent PDE, and the second approach is based on diagonal Padé approximations to (1 + x) − α when the relation of the solution on two times levels is used. The second and fourth order approximations are constructed by both approaches. In order to increase the accuracy of approximations the graded time grid is constructed which compensates the singular behavior of the solution for t close to 0. Results of numerical experiments are presented, they agree well with the theoretical results. [ABSTRACT FROM AUTHOR]

Summary: Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present article, new integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two‐dimensional boundary value problem for fractional diffusion. [ABSTRACT FROM AUTHOR]

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ. The associated Cameron–Martin space is denoted by H. Consider two sufficiently regular convex functions U : X → ℝ and G : X → ℝ. We let ν = e − U μ and Ω = G − 1 (− ∞ , 0 ]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form (ψ , φ) ↦ ∫ Ω ∇ H ψ , ∇ H φ H d ν ψ , φ ∈ W 1 , 2 (Ω , ν) , (0. 1) and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with Ω = X and on half-spaces, namely if U ≡ 0 and G is an affine function, then the domain of the operator defined via (0.1) is the space u ∈ W 2 , 2 (Ω , μ) | ∇ H u (x) , ∇ H G (x) H = 0  for  ρ -a.e.  x ∈ G − 1 (0) , where ρ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure. [ABSTRACT FROM AUTHOR]

In the present work we introduce a weighted Cauchy-Riemann type operator in the complex plane, where the weights are complex non-constant functions. We construct a fundamental solution for this operator where the weights are complex constant functions and orthogonal functions inspired by the idea for the construction of the Levy function proposed by Miranda (see [23]). Therefore we obtain a Cauchy-Pompeiu integral representation formula. We also present some examples of such representations when we take some particular weights. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the following eigenvalue problem of a second order linear elliptic operator: -D\Delta\phi - 2\alpha\nablam(x)\cdot\nabla\phi + V (x)\phi =\lambda\phi in\Omega, complemented by a general boundary condition, including Dirichlet boundary condition and Robin boundary condition,\partial\phi + β(x)φ = O on ∂Ω, where β ∈ C(<9Ω) is allowed to be positive, sign-changing, or negative, and n(x) is the unit exterior normal to ∂Ω at x. The domain Ω C Kλγ is bounded and smooth, the constants D > O and α > O are, respectively, the diffusive and advection coefficients, and m ∈ C2(Ω), V ∈ C(Ω) are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient Í → U or Í → ∞. Our results, together with those of [X. F. Chen and Y. Lou, Indiana Univ. Math. J., 61 (2012), pp. 45--80; A. Devinatz, R. Ellis, and A. Friedman, Indiana Univ. Math. J., 23 (1973/74), pp. 991--1011; and A. Friedman, Indiana U. Math. J., 22 (1973), pp. 1005--1015] where the Neumann boundary case (i.e., β = 0 on ∂Ω) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue. [ABSTRACT FROM AUTHOR]

Given a Lie group G of quantized canonical transformations acting on the space L 2 (M) over a closed manifold M , we define an algebra of so-called G -operators on L 2 (M). We show that to G -operators we can associate symbols in appropriate crossed products with G , introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions on M , transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds. [ABSTRACT FROM AUTHOR]

Abstract Let M be an n (> 2) -dimensional closed orientable submanifold in an (n + p) -dimensional space form R n + p (c). We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by L T f = − div (T ∇ f) , where T is a general symmetric, positive definite and divergence-free (1 , 1) -tensor on M. The upper bound is given in terms of an integration involving tr T and | H T | 2 , where tr T is the trace of the tensor T and H T = ∑ i = 1 n A (T e i , e i) is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous "Reilly inequality". The operator L T can be regarded as a natural generalization of the well-known operator L r which is the linearized operator of the first variation of the (r + 1) -th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method for stable solutions is also presented. [ABSTRACT FROM AUTHOR]

Abstract We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl's asymptotic formula. As an application, we find a lower bound for the mean of the first k eigenvalues of the drifting Laplacian. In particular, we have extended for this operator a partial solution given by Cheng and Yang for the generalized conjecture of Pólya. We also derive a second-Yang type inequality due to Chen and Cheng, and other two inequalities which generalize results by Cheng and Yang obtained for a domain in the unit sphere and for a domain in the projective space. [ABSTRACT FROM AUTHOR]

For a C 2 function u and an elliptic operator L , we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu . An integral version of this estimate is then used to derive a condition for the zero-mean value property of Δ u . An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings. [ABSTRACT FROM AUTHOR]

Let L = - div ⁡ ( A ⁢ ∇ ) {L=-\operatorname{div}(A\nabla)} be a second-order divergence form elliptic operator and let A be an accretive, n × n {n\times n} matrix with bounded measurable complex coefficients in ℝ n {{\mathbb{R}}^{n}}. Let L - α 2 {L^{-\frac{\alpha}{2}}} be the fractional integral associated to L for 0 < α < n {0<\alphak-th order commutator of b and L - α 2 {L^{-\frac{\alpha}{2}}} is given by ( L - α 2 ) b , k ⁢ f ⁢ ( x ) = L - α 2 ⁢ ( ( b ⁢ ( x ) - b ) k ⁢ f ) ⁢ ( x ) . (L^{-\frac{\alpha}{2}})_{b,k}f(x)=L^{-\frac{\alpha}{2}}((b(x)-b)^{k}f)(x). In the paper, we mainly show that if b ∈ BMO ⁢ ( ℝ n ) {b\in\mathrm{BMO}({\mathbb{R}}^{n})} , 0 < λ < n {0<\lambda 0 {\{e^{-tL}\}_{t>0}}. Also, we establish the boundedness of the commutator of the fractional integral with Lipschitz function on Morrey spaces. The results encompass what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. [ABSTRACT FROM AUTHOR]

A time-dependent problem in a bounded domain for a fractional diffusion equation is considered. The first-order evolution equation involves a fractional-power second-order elliptic operator with Robin boundary conditions. A finite-element spatial approximation with an additive approximation of the operator of the problem is used. The time approximation is based on a vector scheme. The transition to a new time level is ensured by solving a sequence of standard elliptic boundary value problems. Numerical results obtained for a two-dimensional model problem are presented. [ABSTRACT FROM AUTHOR]

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients. We also present applications to Dirac operators on Dirac bundles when the metric is non-smooth. [ABSTRACT FROM AUTHOR]

Two control problems with a quadratic cost functional for a parabolic equation with Robin boundary conditions are investigated. [ABSTRACT FROM AUTHOR]

We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns out to be given by the local zeta function of L. In particular, the constant term in the asymptotic expansion of the Green's function $$L^{-1}$$ is often called the mass of L, which (in case that L is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant m-Laplace operators L (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of M is odd and that $$\ker L = 0$$ , and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied. [ABSTRACT FROM AUTHOR]

We study the solvability of the fractional order inhomogeneous Cauchy problem Dαtu(t) = Au(t) + f(t), t > 0, 1 < α ≤ 2; where A is a closed linear operator in some Banach space X and f : [0,∞) → X a given function. Operator families associated with this problem are defined and their regularity properties are investigated. In the case where A is a generator of a β-times integrated cosine family (Cβ(t)), we derive explicit representations of mild and classical solutions of the above problem in terms of the integrated cosine family. We include applications to elliptic operators with Dirichlet, Neumann or Robin type boundary conditions on Lp-spaces and on the space of continuous functions. [ABSTRACT FROM AUTHOR]

In this paper we prove a local Carleman estimate for second order elliptic equations with a general anisotropic Lipschitz coefficients having a jump at an interface. The argument we use is of microlocal nature. Yet, not relying on pseudodifferential calculus, our approach allows one to achieve almost optimal assumptions on the regularity of the coefficients and, consequently, of the interface. [ABSTRACT FROM AUTHOR]

We obtain the asymptotic estimates of the accuracy of eigenvalues of the fourth-order operator with mixed boundary conditions on the boundary of a rectangle. Knowledge of the main part of eigenvalues error allows us to reasonably specify eigenvalues on a sequence of grids, obtain discrete analogs of high accuracy, and construct discrete analogs whose eigenvalues give the bilateral approximation to eigenvalues of the original problem. [ABSTRACT FROM AUTHOR]

Let M n be a compact hypersurface of a real space form and L r the linearized operator of the first variation of the ( r + 1 ) th mean curvature, r ∈ { 0 , 1 ⋯ , n } . In this paper, by a generalized Bochner-type formula for L r and ideas trace back to the Lichnerowicz–Obata Theorem, we obtain some sharp lower bound for the first nonzero eigenvalues of those operators L r . [ABSTRACT FROM AUTHOR]

• Linear scaling selected inverse of elliptic operators. • Fast solver for modified Poisson-Boltzmann equation. • Hierarchical interpolative factorization. This paper studies an efficient numerical method for solving modified Poisson-Boltzmann (MPB) equations with the self Green's function as a state equation to describe electrostatic correlations in ionic systems. Previously, the most expensive point of the MPB solver is the evaluation of Green's function. The evaluation of Green's function requires solving high-dimensional partial differential equations, which is the computational bottleneck for solving MPB equations. Numerically, the MPB solver only requires the evaluation of Green's function as the diagonal part of the inverse of the discrete elliptic differential operator of the Debye-Hückel equation. Therefore, we develop a fast algorithm by a coupling of the selected inversion and hierarchical interpolative factorization. By the interpolative factorization, our new selected inverse algorithm achieves linear scaling to compute the diagonal of the inverse of this discrete operator. The accuracy and efficiency of the proposed algorithm will be demonstrated by extensive numerical results for solving MPB equations. [ABSTRACT FROM AUTHOR]

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. An unsteady problem is considered for a convection and a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem. [ABSTRACT FROM AUTHOR]

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem. [ABSTRACT FROM AUTHOR]

The Schrödinger operator in a thin infinite strip with PT -symmetric boundary conditions and a localized potential is studied. The case of a virtual level on the threshold of the essential spectrum of an efficient one-dimensional operator is considered. Sufficient conditions for the transformation of this level into an isolated eigenvalue are obtained and the first terms of the asymptotic expansion are calculated for this eigenvalue. Sufficient conditions for the absence of such an eigenvalue are also obtained. [ABSTRACT FROM AUTHOR]

Themotivation of this paper is to study a second order elliptic operatorwhich appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant r-mean curvature. We prove a generalizedBochner-type formula for such a kind of operators and as applicationswe obtain some sharp estimates for the first nonzero eigenvalues in two special cases. These results can be considered as generalizations of the Lichnerowicz-Obata Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study complete self-shrinkers in Euclidean space and prove that an n-dimensional complete self-shrinker with polynomial volume growth in Euclidean space Rn+1 is isometric to either Rn, Sn(vn), or Rn-m × Sm(vm), 1 ≤ m ≤ n - 1, if the squared norm S of the second fundamental form is constant and satisfies S < 10/7. [ABSTRACT FROM AUTHOR]

We prove uniformLpestimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3] in the case of Laplace-Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1] very closely. We also show that spectral regions in ourLpresolvent estimates are optimal. [ABSTRACT FROM AUTHOR]

Given V a finite set, a self-adjoint operator K on C ( V ) is called elliptic if it is positive semi-definite and its lowest eigenvalue is simple. Therefore, there exists a unique, up to sign, unitary function ω ∈ C ( V ) satisfying K ( ω ) = λ ω and then, K is named ( λ , ω ) -elliptic . Clearly, a ( λ , ω ) -elliptic operator is singular iff λ = 0 . Examples of elliptic operators are the so-called Schrödinger operators on finite connected networks, as well as the signless Laplacian of connected bipartite networks. A ( λ , ω ) -elliptic operator, K , defines an automorphism on ω ⊥ whose inverse is called orthogonal Green operator of K . We aim here at studying the effect of a perturbation of K on its orthogonal Green operator. The perturbation here considered is performed by adding a self-adjoint and positive semi-definite operator to K . As particular cases we consider the effect of changing the conductances on semi-definite Schödinger operators on finite connected networks and on the signless Laplacian of connected bipartite networks. The expression obtained for the perturbed network is explicitly given in terms of the orthogonal Green function of the original network. [ABSTRACT FROM AUTHOR]

An equation for a fractional power of the second-order elliptic operator is considered. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes are applied. Stability conditions are obtained for the fully discrete schemes under the consideration. The numerical results are presented for a model of two-dimensional problem with a fractional power of an elliptic operator. The dependence of accuracy on grids in time and in space is studied. [ABSTRACT FROM AUTHOR]

We consider divergence form elliptic equations L u : = ∇ ⋅ ( A ∇ u ) = 0 in the half space R + n + 1 : = { ( x , t ) ∈ R n × ( 0 , ∞ ) } , whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A ( x , t ) − A ( x , 0 ) satisfies a Carleson measure condition of Fefferman–Kenig–Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in L p , Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for L , with data in L p ( R n ) , BMO ( R n ) , and C α ( R n ) , and solvability of the Neumann and Regularity problems, with data in the spaces L p ( R n ) / H p ( R n ) and L 1 p ( R n ) / H 1 , p ( R n ) respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials for the t -independent operator L 0 : = − ∇ ⋅ ( A ( ⋅ , 0 ) ∇ ) . [ABSTRACT FROM AUTHOR]

We study in this paper a class of parabolic equations in singular domains. These equations are defined in a singular cylindrical domain whose cross-section contains one reentrant corner or one straight emerging crack. We assume that the diffusion coefficients are non-smooth in the normal direction. We will show some spectral inequality thanks to Carleman type estimates and the construction of a suitable weight function satisfying some properties. As in Benabdallah et al. (C. R. Acad. Sci. Paris 344(6):357-362, ), we deduce the null-controllability of these equations with the help of the Lebeau-Robbiano method. [ABSTRACT FROM AUTHOR]

The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard's example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator $$\hat L$$ or a Volterra well-defined extension of a minimal operator L generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator $$\hat L$$ and of well-defined extensions of the minimal operator L generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra. [ABSTRACT FROM AUTHOR]

Abstract: The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, . The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders. [Copyright &y& Elsevier]