Search

For a fixed bounded Lipschitz domain and a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we investigate a necessary and sufficient condition that an A 1 weight ω must satisfy in order for the weighted W 1 , 2 (ω) estimates for weak solutions of Neumann problems to be true. Moreover, in any Lipschitz domain, under the assumption that the coefficient A is Hölder continuous, we prove that the uniform W 1 , p estimates for solutions to the Neumann problem hold for 2 d d + 1 − δ < p < 2 d d − 1 + δ. As a by-product, in non-periodic setting with A ∈ V M O , we are able to show that the W 1 , p estimates hold for 2 d d + 1 − δ < p < 2 d d − 1 + δ. The ranges are sharp for d = 2 , 3. Finally, we prove an extrapolation result for L p Dirichlet problems for systems of linear elasticity. Specifically, we extrapolate from solvability for 1 < p 0 < 2 (d − 1) d − 2 to the range p 0 < p < 2 (d − 1) d − 2 + δ. The novelty is that the method avoids using the L 2 regularity estimate. [ABSTRACT FROM AUTHOR]

We explore the relationship between rough surface conductors and the phenomenon of passive intermodulation. The underlying surface is taken to be the boundary of a Lipschitz domain, and a characteristic angle of the domain is used to track boundary dependence on the various fields. To model electro‐thermal passive intermodulation in particular, we consider a specific type of temperature‐dependent conductivity and determine conditions on the conductivity under which one can use fixed point arguments to solve an induction heating and Joule heating problem on a Lipschitz domain. In the latter problem, we also consider a time‐dependent permittivity function ε$\varepsilon$. Finally, weak solutions to a magneto‐quasi‐static problem are obtained when the permeability µ is temperature dependent and is allowed to degenerate in a certain way. An interesting effect of the rough surface is the inherently limited Sobolev regularity of the electric field, which can be improved if one assumes additional constraints on the boundary. [ABSTRACT FROM AUTHOR]

We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain Ω subject to homogeneous Dirichlet boundary conditions. We prove L p -resolvent estimates for p satisfying the condition | 1 / p − 1 / 2 | < 1 / 4 + ε for some ε > 0. We further show that the Stokes operator admits the property of maximal regularity and that its H ∞ -calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutions to the Navier–Stokes equations in bounded planar Lipschitz domains. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic stability derived for the two-dimensional incompressible Navier–Stokes equations with multidelays on Lipschitz domain, which models the control theory of 2D fluid flow. By a new retarded Gronwall inequality and estimates of stream function for Stokes equations, the complete trajectories inside pullback attractors are asymptotically stable via the restriction on the generalized Grashof number of fluid flow. The results in this presented paper are some extension of the literature by Yang, Wang, Yan and Miranville in 2021, as well as also the preprint by Su, Yang, Miranville and Yang in 2022 [ABSTRACT FROM AUTHOR]

We obtain a system of boundary-domain integral equations (BDIE) equivalent to the Dirichlet problem for the diffusion equation in non-homogeneous media. We use an extended version of the boundary integral method for PDEs with variable coefficients for which a parametrix is required. We generalize existing results for this family of parametrices considering a non-smooth variable coefficient in the PDE and source term in Hs-2(Ω), 1/2< s <3/2 defined on a Lipschitz domain. The main results concern the equivalence between the original BVP and the corresponding BDIE system, as well as the well-posedness of the BDIE system. [ABSTRACT FROM AUTHOR]

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L : = ∇ X ⋅ (A (X , Y , t) ∇ X) + X ⋅ ∇ Y − ∂ t , in unbounded domains of the form Ω = { (X , Y , t) = (x , x m , y , y m , t) ∈ R m − 1 × R × R m − 1 × R × R | x m > ψ (x , y , y m , t) } , where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L. Concerning A = A (X , Y , t) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in R m). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded domain in R n with C 1 boundary and let u λ be a Dirichlet Laplace eigenfunction in Ω with eigenvalue λ . We show that the (n - 1) -dimensional Hausdorff measure of the zero set of u λ does not exceed C (Ω) λ . This result is new even for the case of domains with C ∞ -smooth boundary. [ABSTRACT FROM AUTHOR]

The interior Dirichlet boundary value problem for the diffusion equation in nonhomogeneous media is reduced to a system of boundary‐domain integral equations (BDIEs) employing the parametrix obtained in Portillo (2019). We further extend the results obtained in Portillo (2019) for the mixed problem in a smooth domain with L2(Ω) right‐hand side to Lipschitz domains and partial differential equation (PDE) right‐hand side in the Sobolev space H−1(Ω), where neither the classical nor the canonical conormal derivatives are well‐defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω be a bounded Lipschitz domain in R n. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p ∈ (2 , ∞) , two necessary and sufficient conditions for W 1 , p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1 , q estimates of solutions with q ∈ [ 2 , p ] and some Muckenhoupt weights, are obtained. As applications, for any given p ∈ (1 , ∞) and ω ∈ A p (R n) (the class of Muckenhoupt weights), the authors establish weighted W ω 1 , p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces. [ABSTRACT FROM AUTHOR]

We use the method of layer potentials to study the R2 regularity problem and the D2 Dirichlet problem for second order elliptic equations with lower order coefficients in bounded Lipschitz domains. For R2 we establish existence and uniqueness by assuming that L is of the form Lu = -div(A∇u+ bu)+c∇u+du, where the matrix A is uniformly elliptic and Hölder continuous, b is Hölder continuous, and c, d belong to Lebesgue classes and satisfy either the condition d ≥ div b or d ≥ div c in the sense of distributions. In particular, A is not assumed to be symmetric, and there is no smallness assumption on the norms of the lower order coefficients. We also show existence and uniqueness for D2 for the adjoint equations Ltu = 0. [ABSTRACT FROM AUTHOR]

We construct a bounded C1 domain Ω in Rn for which the H3/2 regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in C∞(Ω¯) such that the solution of Δu=f in Ω and either u=0 on ∂Ω or ∂nu=0 on ∂Ω is contained in H3/2(Ω) but not in H3/2+ε(Ω) for any ε>0. An analogous result holds for Lp Sobolev spaces with p∈(1,∞). [ABSTRACT FROM AUTHOR]

Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the Lp Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in ℝd is solvable for some 1 < p = p 0 < 2 (d − 1) d − 2 , then it is solvable for all p satisfying p 0 < p < 2 (d − 1) d − 2 + ε. The proof is based on a real-variable argument. It only requires that local solutions of L(u) = 0 satisfy a boundary Cacciopoli inequality. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ Cn be a bounded Lipschitz q-pseudoconvex domain that admit good weight functions. We shall prove that the canonical solution operator for the @̄-equation is compact on the boundary of and is bounded in the Sobolev space Wkr;s(Ω) for some values of k. Moreover, we show that the Bergman projection and the @̄-Neumann operator are bounded in the Sobolev space Wkr,s(Ω) for some values of k. If Ω is smooth, we shall give suffcient conditions for compactness of the @̄-Neumann operator. [ABSTRACT FROM AUTHOR]

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) H 1 norm of the solution is bounded by the L 2 norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature, while here we assume only Lipschitz regularity and star-shapedness with respect to a point. Furthermore, our bounds are obtained using identities first introduced by Morawetz (essentially integration by parts), whereas the existing bounds use the much-more sophisticated technology of microlocal analysis and propagation of singularities. We also adapt existing results to show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible. [ABSTRACT FROM AUTHOR]

In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the L ∞ (H 1) and L ∞ (L 2) norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed. [ABSTRACT FROM AUTHOR]

In the paper, we establish commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains. The approach is based on Dahlberg's bilinear estimates, and the results may be regarded as an extension of [8,21] to Stokes systems. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 and Ω be a bounded Lipschitz domain in R n . Assume that p ∈ ( 2 , ∞ ) and the function b ∈ L ∞ ( ∂ Ω ) is non-negative, where ∂Ω denotes the boundary of Ω. Denote by ν the outward unit normal to ∂Ω. In this article, the authors give two necessary and sufficient conditions for the unique solvability of the Robin problem for the Laplace equation Δ u = 0 in Ω with boundary data ∂ u / ∂ ν + b u = f ∈ L p ( ∂ Ω ) , respectively, in terms of a weak reverse Hölder inequality with exponent p or the unique solvability of the Robin problem with boundary data in some weighted L 2 ( ∂ Ω ) space. As applications, the authors obtain the unique solvability of the Robin problem for the Laplace equation in the bounded (semi-)convex domain Ω with boundary data in (weighted) L p ( ∂ Ω ) for any given p ∈ ( 1 , ∞ ) . [ABSTRACT FROM AUTHOR]

Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others. [ABSTRACT FROM AUTHOR]

We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio. [ABSTRACT FROM AUTHOR]

The paper extends the results obtained by Kenig, Lin, and Shen [Arch. Ration. Mech. Anal., 203 (2012), pp. 1009--1036] to more general elliptic homogenization problems in two perspectives: lower order terms in the operator and no smoothness on th e coefficients. We do not repeat their arguments. Instead we find the new weighted-type estimates for the smoothing operator at scale e, and combining some techniques developed by Shen in [preprint, arXiv:1505.00694v1, 2015] leads to our main results. In addition, we also obtain sharp O(ε) convergence rates in Lp with p = 2d/(d - 1), which were originally established by Shen for elasticity systems in [preprint, arXiv:1505.00694v1, 2015]. Also, this work may be regarded as the extension of [T. Suslina, Mathematika, 59 (2013), pp. 463-476; T. Suslina SIAM J. Math. Anal, 45 (2013), pp. 3453-3493] developed by Suslina concerned with the bounded Lipschitz domain. [ABSTRACT FROM AUTHOR]

This paper is devoted to classical spectral boundary value problems for strongly elliptic second-order systems in bounded Lipschitz domains, in general non-self-adjoint, namely, to questions of regularity and completeness of root functions (generalized eigenfunctions), resolvent estimates, and summability of Fourier series with respect to the root functions by the Abel-Lidskii method in Sobolev-type spaces. These questions are not difficult in the Hilbert spaces of the type [ABSTRACT FROM AUTHOR]

We introduce an arbitrary order, computationally efficient method to smooth corners on curves in the plane, as well as edges and vertices on surfaces in R3: The method is local, only modifying the original surface in a neighborhood of the geometric singularity, and preserves desirable features like convexity and symmetry. The smoothness of the final surface is an explicit parameter in the method, and the band-limit of the smoothed surface is proportional to its smoothness. Several numerical examples are provided in the context of acoustic scattering. In particular, we compare scattered fields from smoothed geometries in two dimensions with those from polygonal domains. We observe that significant reductions in computational cost can be obtained if merely approximate solutions are desired in the near-or far-field. Provided that the smoothing is subwavelength, the error of the scattered field is proportional to the size of the geometry that is modified. [ABSTRACT FROM AUTHOR]

The square root of the heat operator ∂ t − Δ , can be realized as the Dirichlet to Neumann map of the heat extension of data on R n + 1 to R + n + 2 . In this note we obtain similar characterizations for general fractional powers of the heat operator, ( ∂ t − Δ ) s , s ∈ ( 0 , 1 ) . Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem. [ABSTRACT FROM AUTHOR]

We examine the asymptotic behaviour of u ˙ = d A u + f ( u ) for positive initial values u ( 0 ) > 0 . Here A is the generator of an exponentially bounded semigroup on L ∞ ( Ω ) with several additional properties. A particular example is the Laplace operator Δ Ω R , ∞ with Robin boundary conditions on a Lipschitz domain defined on L ∞ ( Ω ) . Moreover, f : L ∞ ( Ω ) → L ∞ ( Ω ) is a local, continuously Fréchet-differentiable, and strictly concave map with f ( 0 ) = 0 . We will analyse the asymptotic behaviour of the solutions as t → ∞ and its dependency on the diffusion coefficient d > 0 . [ABSTRACT FROM AUTHOR]

The purpose of this work is to show the well-posedness in L2-Sobolev spaces of the Poisson-transmission problem for the Oseen and Brinkman systems on complementary Lipschitz domains in a compact Riemannian manifold. The Oseen system appears as a perturbation of order one of the Stokes system, given in terms of the Levi-Civita connection, while the Brinkman system is a zero order perturbation of the Stokes system. The technical details of this paper rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by a first order differential operator given in terms of the Levi-Civita connection. The compactness of this differential operator requires to restrict ourselves to low dimensional compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Under suitable hypotheses on the continuous delay, distributed delay, and the initial data in this paper, the large-time behavior for the 2D Navier-Stokes-Voigt equations with continuous delay and distributed delay on the Lipschitz domain is studied. The existence of pullback attractors in the non-smooth domain was obtained via verifying some pullback dissipation and asymptotical compactness for the continuous process. [ABSTRACT FROM AUTHOR]

Under suitable hypotheses on the continuous delay, distributed delay, and the initial data in this paper, the large-time behavior for the 2D Navier-Stokes-Voigt equations with continuous delay and distributed delay on the Lipschitz domain is studied. The existence of pullback attractors in the non-smooth domain was obtained via verifying some pullback dissipation and asymptotical compactness for the continuous process. [ABSTRACT FROM AUTHOR]

The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded Lipschitz domains. [ABSTRACT FROM AUTHOR]

The purpose of this work is to show the well-posedness in $$L^2$$ -Sobolev spaces for a Poisson-transmission problem involving $$L^{\infty }$$ -perturbations of the Stokes system on complementary Lipschitz domains in compact Riemannian manifolds. The technical details rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by suitable compact operators. The compact part is provided by the $$L^{\infty }$$ -perturbations of the Stokes system. An existence result for a related semilinear Poisson-transmission problem is also formulated. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy- Forchheimer-Brinkman system in a bounded Lipschitz domain in Rn(n = 2, 3), with small boundary datum in L2-based Sobolev spaces. A useful intermediary result is thewell-posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in Rn(n ⩾ 2), with Dirichlet boundary condition and data in L2-based Sobolev spaces. We obtain this well-posedness result by showing that thematrix type operator associated with the Poisson problem is an isomorphism. Then,we combine thewell-posedness result from the linear case with a fixed point theoremin order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy-Forchheimer-Brinkman system. Some applications are also included. [ABSTRACT FROM AUTHOR]

In this paper we consider scalar parabolic equations in a general non-smooth setting emphasizing interface conditions and mixed boundary conditions. In particular, we study dynamics and diffusion on a Lipschitz interface and on the boundary, where the diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we consider diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal L p -regularity and bounded H ∞ -calculus for the corresponding operator, providing well-posedness for a large class of initial conditions and external forces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to combine a layer potential analysis with the Schauder fixed point theorem to show the existence of solutions of the Poisson problem for a semilinear Brinkman system on bounded Lipschitz domains in $${{\mathbb R}^n (n\geq 2)}$$ with Dirichlet or Robin boundary conditions and data in L-based Sobolev spaces. We also obtain an existence and uniqueness result for the Dirichlet problem for a special semilinear elliptic system, called the Darcy-Forchheimer-Brinkman system. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi-boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kreĭn type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results by Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish the L p resolvent estimates on a Lipschitz domain Ω in R d for constant coefficient elliptic systems with homogeneous Neumann boundary conditions, where 1 < p < ∞ for d = 3 , and 2 d / ( d + 3 ) − ϵ < p < 2 d / ( d − 3 ) + ϵ for d ≥ 4 with some positive constant ϵ = ϵ ( Ω ) . We also give the global L p estimates for the derivatives of solutions to the previous systems, where 2 ≤ p < 2 d / ( d − 1 ) + ϵ for d ≥ 3 . Finally we extend our main results to the case of some variable coefficient elliptic systems on a bounded Lipschitz domain. [ABSTRACT FROM AUTHOR]

We investigate Schrödinger operators with δ- and δ′-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result, we prove an operator inequality for the Schrödinger operators with δ- and δ′-interactions which is based on an optimal coloring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ′-interactions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study, for given p, , the boundary behavior of non-negative solutions to equations of p-Laplace type of the form Concerning the matrix A we assume that A is symmetric, positive definite and that its entries are -smooth functions. Let be an Ahlfors regular NTA-domain with constants , and let σ be the surface measure on ∂Ω. Given , , suppose that u is a positive solution to in , that u is continuous in and that on . Among other things we prove that , for σ-a.e. , as non-tangentially in Ω, and that where are the constants giving the structural restrictions on our class of operators. Furthermore, assuming in addition that is uniformly δ-approximable by Lipschitz graphs, we prove Hölder continuity of the ratio of two solutions which vanish on a portion of the boundary. Our main contribution is the extension of a toolbox developed by the second author and John Lewis in the context of p-harmonic functions, to non-negative solutions to the equation in Lipschitz domains. [Copyright &y& Elsevier]

We establish fractional Hardy-type inequalities in a bounded domain with uniformly fat complement. In particular, our results apply in bounded C∞ domains and Lipschitz domains. [ABSTRACT FROM AUTHOR]

Abstract: We consider a Sturm–Liouville boundary value problem in a bounded domain of . By this is meant that the differential equation is given by a second order elliptic operator of divergent form in and the boundary conditions are of Robin type on . The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact selfadjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types. [Copyright &y& Elsevier]

In this paper we obtain the existence and uniqueness in some Sobolev and Lp spaces for a Dirichlet-transmission problem for the Stokes and Brinkman equations on Lipschitz domains in ℝn, n > 3. In the particular case n = 3 this problem describes the Stokes flow of a viscous incompressible fluid past a porous particle and in presence of a solid core. The flow within the porous particle is described by the Brinkman equation. In order to obtain the desired existence and uniqueness result, we use an indirect boundary integral formulation based on the layer potential theory for both Brinkman and Stokes equations. Some special cases, which refer to the Stokes flow past a porous particle with large permeability, respectively low permeability, are also analyzed. [ABSTRACT FROM AUTHOR]

We establish a sharp regularity result for the normal trace of the solution to the anisotropic linear elasticity system with Dirichlet boundary condition on a Lipschitz domain. Using this result we obtain a new existence result for a fluid-structure interaction model in the case when the structure is an anisotropic elastic body. [ABSTRACT FROM AUTHOR]

Abstract: Let be a bounded domain, or a Lipschitz domain “flat enough”, and consider the Beurling transform of : Using a priori estimates, in this paper we solve the following free boundary problem: if belongs to the Sobolev space for , such that , then the outward unit normal N on ∂Ω is in the Besov space . The converse statement, proved previously by Cruz and Tolsa, also holds. So we have Together with recent results by Cruz, Mateu and Orobitg, from the preceding equivalence one infers that the Beurling transform is bounded in if and only if the outward unit normal N belongs to , assuming that . [Copyright &y& Elsevier]

Abstract: An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials. [Copyright &y& Elsevier]

Let Ω be a bounded Lipschitz domain in ℝ, n ⩽ 2, and let L be a second-order matrix strongly elliptic operator in Ω written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation Lu = f, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well. We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary Γ = ∂Ω or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem. [ABSTRACT FROM AUTHOR]

We prove pointwise estimates for the heat kernel of a second-order elliptic operator with Dirichlet boundary conditions on a bounded Lipschitz domain in ℝn, n ≥ 1. Applications to obtain estimates for intrinsic ultracontractivity of the heat semigroup and expected exit time of a Brownian motion are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a variational method for a multi-dimensional inverse heat conduction problem in Lipschitz domains. We regularize the problem by using the boundary element method coupled with the conjugate gradient method. We prove the convergence of this scheme with and without Tikhonov regularization. Numerical examples are given to show the efficiency of the scheme. [ABSTRACT FROM AUTHOR]

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated. [ABSTRACT FROM AUTHOR]

Abstract: Let be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform estimate in a Lipschitz domain Ω in for solutions to the Dirichlet problem: in Ω and on ∂Ω, where and C, are constants independent of . The ranges are sharp for or 3. [Copyright &y& Elsevier]

Abstract: In this paper we continue the study in Lewis and Nyström (2010) , concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph. [Copyright &y& Elsevier]

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H, the simplest L-spaces of the Sobolev type, with the use of potential type operators on S. We also discuss, first, the regularity of solutions in somewhat more general Bessel potential spaces and Besov spaces and, second, the spectral properties of problems with spectral parameter in the transmission conditions on S, including the asymptotics of the eigenvalues. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are -smooth for some . As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain. [ABSTRACT FROM AUTHOR]