The variational solution to the Zaremba problem for a divergent linear second order elliptic equation with measurable coefficients is considered. The problem is set in a local Lipschitz graph domain. An estimate in L_{2+δ}, δ > 0, for the gradient of a solution, is proved. An example of the problem with the Dirichlet data supported by a fractal set of zero (n - 1)-dimensional measure and non-zero p-capacity, p > 1 is constructed.
We discuss a sharp generalization of the Hopf-Oleinik boundary point principle (BPP) for domains satisfying an interior pseudo-ball condition, for non-divergence form, semi-elliptic operators with singular drift. In turn, this result is used to derive a version of the strong maximum principle under optimal pointwise blow-up conditions for the coefficients of the differential operator involved. We also explain how a uniform two-sided pseudo-ball condition may be used to provide a purely geometric characterization of Lyapunov domains, and clarify the role this class of domains plays vis-a-vis to the BPP.
Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented. (C) 2010 Elsevier Masson SAS. All rights reserved.
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in R-n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.
Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.
The existence of solutions to a class of quasilinear elliptic problems on noncompact Riemannian manifolds, with finite volume, is investigated. Boundary value problems, with homogeneous Neumann conditions, in possibly irregular Euclidean domains are included as a special instance. A nontrivial solution is shown to exist under an unconventional growth condition on the right-hand side, which depends on the geometry of the underlying manifold. The identification of the critical growth is a crucial step in our analysis, and entails the use of the isocapacitary function of the manifold. A condition involving its isoperimetric function is also provided. (C) 2017 Elsevier Inc. All rights reserved.
We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)del u(x)) = 0 in an n-dimensional domain Omega, n greater than or equal to 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain Omega, which in turn lead to a sufficient condition for the Holder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmen-Lindelof type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div(A(.)del u(.)) in a domain Omega and for the solutions of the nonhomogeneous equation -div(A(x)del u(x)) = mu with measure on the right-hand side.
We initiate the study of the finiteness condition∫ Ω u(x) −β dx≤C(Ω,β)<+∞ whereΩ⊆R n is an open set and u is the solution of the Saint Venant problem Δu=−1 in Ω , u=0 on ∂Ω . The central issue which we address is that of determining the range of values of the parameter β>0 for which the aforementioned condition holds under various hypotheses on the smoothness of Ω and demands on the nature of the constant C(Ω,β) . Classes of domains for which our analysis applies include bounded piecewise C 1 domains in R n , n≥2 , with conical singularities (in particular polygonal domains in the plane), polyhedra in R 3 , and bounded domains which are locally of classC 2 and which have (finitely many) outwardly pointing cusps. For example, we show that if u N is the solution of the Saint Venant problem in the regular polygon Ω N with N sides circumscribed by the unit disc in the plane, then for each β∈(0,1) the following asymptotic formula holds: % {eqnarray*} \int_{\Omega_N}u_N(x)^{-\beta}\,dx=\frac{4^\beta\pi}{1-\beta} +{\mathcal{O}}(N^{\beta-1})\quad{as}\,\,N\to\infty. {eqnarray*} % One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions v satisfying v(0)=0 , ∇v(0)=0 and Δv≥c>0 .
Various notions of dissipativity for partial differential operators and their applications are surveyed. We deal with functional dissipativity and its particular case L-p-dissipativity. Most of the results are due to the authors.
In the present paper we consider the Dirichlet problem for the second order differential operator E = del(A del), where A is a matrix with complex valued L-infinity entries. We introduce the concept of dissipativity of E with respect to a given function phi : R+ -> R+. Under the assumption that the ImA is symmetric, we prove that the condition vertical bar s phi (s)vertical bar vertical bar < ImA (x)xi, xi >vertical bar <= 2 root phi(s)[s phi(s)] < ReA (x)xi, xi > (for almost every x is an element of Omega subset of R-N and for any s > 0, xi is an element of R-N) is necessary and sufficient for the functional dissipativity of E. (c) 2020 Elsevier Ltd. All rights reserved.
After introducing the concept of functional dissipativity of the Dirichlet problem in a domain Ω⊂RN for systems of partial differential operators of the form ∂h(Ahk(x)∂k) (Ahk(x) being m×m matrices with complex valued L∞ entries), we find necessary and sufficient conditions for the functional dissipativity of the two-dimensional Lamé system. As an application of our theory we provide two regularity results for the displacement vector in the N-dimensional equilibrium problem, when the body is fixed along its boundary.
We find necessary and sufficient conditions for the L-p-dissipatiyity of the Dirichlet problem for systems of partial differential operators of the first order with complex locally integrable coefficients. As a by product we obtain sufficient conditions for a certain class of systems of the second order.
A priori bounds for solutions to (nonlinear) elliptic Neumann problems in open subsets O of Rn are established via inequalities relating the Lebesgue measure of subsets of O to their relative capacity. Both norm and capacitary estimates for solutions, and norm estimates for their gradients are derived which improve classical results even in the case of the Laplace equation. © 2007 Elsevier Masson SAS. All rights reserved.
We deal with Neumann problems for Schrodinger type equations, with non-necessarily bounded potentials, in possibly irregular domains in R-n. Sharp balance conditions between the regularity of the domain and the integrability of the potential for any solution to be bounded are established. The regularity of the domain is described either through its isoperimetric function or its isocapacitary function. The integrability of the sole negative part of the potential plays a role, and is prescribed via its distribution function. The relevant conditions amount to the membership of the negative part of the potential to a Lorentz type space defined either in terms of the isoperimetric function, or of the isocapacitary function of the domain. (c) 2012 Elsevier Masson SAS. All rights reserved.
We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds M of finite volume. Sharp conditions ensuring L-q(M) and L-infinity(M) bounds for eigenfunctions are exhibited in terms of either the isoperimetric function or the isocapacitary function of M.
Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.
This is a survey of some recent contributions by the authors on global integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Minimal assumptions on the regularity of the ground domain and of the prescribed data are pursued.
A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.
Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold M are established in terms of the isocapacitary function of M. The relevant capacity takes a different form according to whether M has finite or infinite volume. Conditions involving the more standard isoperimetric function of M can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples.
Second-order estimates are established for solutions to the p-Laplace system with right-hand side in L-2. The nonlinear expression of the gradient under the divergence operator is shown to belong to W-1,W-2, and hence to enjoy the best possible degree of regularity. Moreover, its norm in 1471,2 is proved to be equivalent to the norm of the right-hand side in L-2. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local estimates for local solutions are provided as well. (C) 2019 Elsevier Masson SAS. All rights reserved.
Boundary value problems for a class of quasilinear elliptic equations, with an Orlicz type growth and L-1 right-hand side are considered. Both Dirichlet and Neumann problems are contemplated. Existence and uniqueness of generalized solutions, as well as their regularity, are established. The case of measure right-hand sides is also analyzed. (C) 2017 Published by Elsevier Ltd.
Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.
Best possible second-order regularity is established for solutions to p-Laplacian type equations with and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L (2)-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all.
Sobolev embeddings into Orlicz spaces on domains in the Eu-clidean space or, more generally, on Riemannian manifolds are considered. Highly irregular domains where the optimal degree of integrability of a func-tion may be lower than the one of its gradient are focused. A necessary and sufficient condition for the validity of the relevant embeddings is established in terms of the isocapacitary function of the domain. Compact embeddings are discussed as well. Sufficient conditions involving the isoperimetric function of the domain are derived as a by-product.
A form of Sobolev inequalities for the symmetric gradient of vector-valued functions is proposed, which allows for arbitrary ground domains in Rn. In the relevant inequalities, boundary regularity of domains is replaced with information on boundary traces of trial functions. The inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of Sobolev functions, in regular domains. Furthermore, they involve constants independent of the geometry of the domain, and hence yield novel results yet for smooth domains. Our approach relies upon a pointwise estimate for the functions in question via a Riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace. (C) 2019 Elsevier Ltd. All rights reserved.
A theory of Sobolev inequalities in arbitrary open sets in R-n is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities exhibit the same critical exponents as in the classical framework. Moreover, they involve constants independent of the geometry of the domain, and hence yield genuinely new results even in the case when just smooth domains are considered. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. (C) 2016 Elsevier Inc. All rights reserved.
The Lipschitz continuity of solutions to Dirichlet and Neumann problems for nonlinear elliptic equations, including the p-Laplace equation, is established under minimal integrability assumptions on the data and on the curvature of the boundary of the domain. The case of arbitrary bounded convex domains is also included. The results have new consequences even for the Laplacian.
We show that the sharp constant in the classical n-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for n = 2 without the axisymmetry assumption.
Let Omega be a smooth bounded domain in R-N, N greater than or equal to 3. We show that Hardy's inequality involving the distance to the boundary, with best constant (1/4), may still be improved by adding a multiple of the critical Sobolev norm.
Motivated by a question of Brezis and Marcus, we show that the L^{p}–Hardy inequality involving the distance to the boundary of a convex domain, can be improved by adding an L^{q} norm q ≥ p, with a constant depending on the interior diameter of Ω.
It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Mazya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).
Asymptotic approximations for Greens function for the operator -Delta in a long rod are derived. These approximations are uniformly valid over the whole domain including the end regions of the rod. Connections are established between the asymptotic approximations in a long rod and the asymptotics in thin domains. Overview of asymptotic approximations of Greens kernels in a domain with a small hole and domains with singularly perturbed smooth or conical boundaries is also given.
The authors prove a maximum modulus estimate for solutions of the stationary Navier-Stokes system in bounded domains of polyhedral type.
The three-dimensional (3D) Navier-Stokes equations less thanbrgreater than less thanbrgreater thanu(t) + (u . del)u = -del p + Delta u, divu = 0 in Q(0), (0.1) less thanbrgreater than less thanbrgreater thanwhere u = [u, v, w](T) is the vector field and p is the pressure, are considered. Here, Q(0) subset of R-3 x [-1, 0) is a smooth domain of a typical backward paraboloid shape, with the vertex (0, 0) being its only characteristic point: the plane {t = 0} is tangent to. partial derivative Q(0) at the origin, and other characteristics for t is an element of [0,-1) intersect. partial derivative Q(0) transversely. Dirichlet boundary conditions on the lateral boundary. partial derivative Q(0) and smooth initial data are prescribed: less thanbrgreater than less thanbrgreater thanu = 0 on. partial derivative Q(0), and u(x, -1) = u(0)(x) in less thanbrgreater than less thanbrgreater thanQ(0) boolean AND {t = -1} (div u(0) = 0). (0.2) less thanbrgreater than less thanbrgreater thanExistence, uniqueness, and regularity studies of (0.1) in non-cylindrical domains were initiated in the 1960s in pioneering works by Lions, Sather, Ladyzhenskaya, and Fujita-Sauer. However, the problem of a characteristic vertex regularity remained open. less thanbrgreater than less thanbrgreater thanIn this paper, the classic problem of regularity (in Wieners sense) of the vertex (0, 0) for (0.1), (0.2) is considered. Petrovskiis famous "2 root log log-criterion of boundary regularity for the heat equation (1934) is shown to apply. Namely, after a blow-up scaling and a special matching with a boundary layer near. partial derivative Q(0), the regularity problem reduces to a 3D perturbed nonlinear dynamical system for the first Fourier-type coefficients of the solutions expanded using solenoidal Hermite polynomials. Finally, this confirms that the nonlinear convection term gets an exponentially decaying factor and is then negligible. Therefore, the regularity of the vertex is entirely dependent on the linear terms and hence remains the same for Stokes and purely parabolic problems. less thanbrgreater than less thanbrgreater thanWell-posed Burnett equations with the minus bi-Laplacian in (0.1) are also discussed.
Parameters of moving sources on the sea surface are recovered by spectral analysis of the induced surface waves. The method can be an alternative to the standard way of seeing the ship directly, in particular, when the direct observation is impossible. © 2003 Elsevier B.V. All rights reserved.
We introduce a class of weak solutions to the quasilinear equation -Delta(p)u = sigma vertical bar u vertical bar(P-2)u in an open set Omega subset of R-n with p andgt; 1, where Delta(p)u = del. (vertical bar del u vertical bar(p-2)del u) is the p-Laplacian operator. Our notion of solution is tailored to general distributional coefficients sigma which satisfy the inequality less thanbrgreater than less thanbrgreater than-Lambda integral(Omega) vertical bar del h vertical bar(p)dx andlt;= andlt;vertical bar h vertical bar(p), sigma andgt; andlt;= lambda integral(Omega) vertical bar del h vertical bar(p)dx, less thanbrgreater than less thanbrgreater thanfor all h is an element of C-0(infinity)(Omega). Here 0 andlt; Lambda andlt; +infinity-, and less thanbrgreater than less thanbrgreater than0 andlt; lambda andlt; (p - 1)(2-p) if p andgt;= 2, or 0 andlt; lambda andlt; 1 if 0 andlt; p andlt; 2. less thanbrgreater than less thanbrgreater thanAs we shall demonstrate, these conditions on lambda are natural for the existence of positive solutions, and cannot be relaxed in general. Furthermore, our class of solutions possesses the optimal local Sobolev regularity available under such a mild restriction on sigma. less thanbrgreater than less thanbrgreater thanWe also study weak solutions of the closely related equation -Delta p nu = (p - 1)vertical bar del nu vertical bar(p) +sigma, under the same conditions on . Our results for this latter equation will allow us to characterize the class of sigma satisfying the above inequality for positive lambda and Lambda. thereby extending earlier results on the form boundedness problem for the Schrodinger operator to p not equal 2.
We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schrödinger type, for an arbitrary open Ω ⊆ ℝn under only a form-boundedness assumption on σ ∈ D′(Ω) and ellipticity assumption on A ∈ L∞(Ω)n×n. We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient, As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schrödinger type operator H = -div(A∇·)-σ with arbitrary distributional potential σ ∈ D′(Ω), and give examples clarifying the relationship between these two properties. © 2012 Hebrew University Magnes Press.
We construct a function u is an element of W-loc(1.1) (B(0, 1)) which is a solution to div(A del u) = 0 in the sense of distributions, where A is continuous and u is not an element of W-loc(1.p) (B(0, 1)) for p greater than 1. We also give a function u is an element of W-loc(1.1)(B(0, 1)) such that u is an element of W-loc(1.p) (B(0, 1)) for every p less than infinity, u satisfies div(A del u) = 0 with A continuous but u is not an element of W-loc(1.infinity) (B(0, 1)). This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335-338).
We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a submerged body. Under some geometrical requirements, we derive an explicit bound for the solution depending on the domain and the functions on the right-hand side.
We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a critically submerged body (i.e., the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyze its properties. Under a geometric condition introduced by V. Mazya in 1978, we prove an important property of the scattering matrix, which may be interpreted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g., H-loc(2) boolean AND L-infinity) without use of the radiation conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The fact that the existence and uniqueness result does not rely on either the radiation conditions or the limiting absorption principle is the first result of this type known to us in the theory of linear wave problems in unbounded domains.
In this paper we analyze hypersingular integral equations of the Peierls type specified on the whole real axis. The method of "approximate approximations" [Maz'ya, Proceedings of the Eighth Conference on the Mathematics of Finite Elements and Applications, VIII, MAFELAP 1993, Brunel University, 1994, pp. 107-104] is employed in order to produce numerical solutions of high accuracy, and the results are applied to problems in the theory of dislocations. The critical Peierls stress is evaluated for different configurations of dislocations and for different types of the interatomic force law.
The aim of this work is the accurate calculation of periodic solutions to the Sivashinsky equation, which models dynamics of the long wave instability of laminar premixed flame. A highly accurate computational algorithm was developed in both one and two spatial dimensions and its crucial implementation details are presented. The algorithm is based on the concept of saturated asymptotic approximations and can be straightforwardly extended to a wide variety of nonlinear integro-differential equations. The development of such an algorithm was motivated by difficulties in interpretation of the results of numerical experiments with the Sivashinsky equation using spectral methods. The computations carried out by the algorithm in question are in good agreement with the results obtained earlier by spectral methods. Analysis of the accuracy of obtained numerical solutions and of their stabilization to steady states supports the idea of the instability of the steady coalescent pole solutions (with maximal possible number of poles) to the Sivashinsky equation in large domains through huge linear transient amplification of nonmodal perturbations of small but finite amplitudes.
We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrodinger operators with a positive scalar potential. They are expressed in terms of Wiener's capacity and the local energy of the magnetic field. The conditiions for the discreteness of spectrum depend, in particular, on a functional parameter which is a decreasing function of one variable whose argument is the normalized local energy of the magnetic field. This function enters the negligibility condition of sets for the scalar potential. We give a description for the range of all admissible functions which is precise in a certain sense. In case when there is no magnetic field, our results extend the discreteness of spectrum and positivity criteria by Molchanov [Molchanov, A. M. (1953). On the discreteness of the spectrum conditions for self-adjoint differential equations of the second order (Russian). Trudy Mosk. Matem. Obshchestva (Proc. Moscow Math. Society) 2:169-199] and Maz'ya [Maz'ya, V. G. (1973). On (p, l)-capacity, imbedding theorems and the spectrum of a self-adjoint elliptic operator. Math. USSR Izv. 7:357-387].
[No abstract available]