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1.
We study the initial-boundary-value problems for multidimensional scalar conservation laws in noncylindrical domains with Lipschitz boundary. We show the existence-uniqueness of this problem for initial-boundary data in L and the flux-function in the class C1. In fact, first considering smooth boundary, we obtain the L1-contraction property, discuss the existence problem and prove it by the Young measures theory. In the end we show how to pass the existence-uniqueness results on to some domains with Lipschitz boundary.  相似文献   

2.
For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt–Caffarelli–Friedman and Caffarelli–Jerison–Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace–Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.  相似文献   

3.
The Calderón Projector, is one of the most important tools in the study of boundary value problems for elliptic operators. Its construction is well known for elliptic operators with C coefficients on C domains and even for the Laplacian operator on C1 domains. The aim of this article is to extend the results for the Laplacian case to elliptic operators in divergence form with Lipschitz coefficients on C1 domains.  相似文献   

4.
In this paper we discuss the solvability of boundary value problems for the Laplace operator on Lipschitz domains with arbitrary topology via boundary layers. An application to hydrodynamics is included.Partially supported by a UMC Research Board grant and UMC Summer Research Fellowship  相似文献   

5.
We study a two-phase free boundary problem in which the speed of the free boundary depends also on its curvature. It is assumed that the free boundary is Lipschitz and it is proved that the solution as well as the free boundary are classical.  相似文献   

6.
Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.  相似文献   

7.
In this paper we continue the study in Lewis and Nyström (2010) [19], 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.  相似文献   

8.
We investigate a class of quasi-linear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded non-smooth domains, which may include non-Lipschitz domains, such as domains with fractal boundary and rough domains. We obtain solvability and global regularity results for both the elliptic and parabolic Robin problem. Some a priori estimates, as well as fine properties for the corresponding nonlinear semigroups, are established. As a consequence, we generalize the global regularity theory for the Robin problem over non-smooth domains by extending it for the first time to the variable exponent case, and furthermore, to the anisotropic variable exponent case.  相似文献   

9.
We establish sharp regularity and Fredholm theorems for the operator on domains satisfying some nongeneric geometric conditions. We use these domains to construct explicit examples of bad behavior of the Kohn Laplacian: It is not always hypoelliptic up to the boundary, its partial inverse is not compact and it is not globally subelliptic.  相似文献   

10.
We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the full b‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems.  相似文献   

11.
B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in Lp spaces p > 2 ? δ are extended to compact polyhedral domains of ?n. Consequently, for q < 2 + δ, Dahlberg’s reverse Hölder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.  相似文献   

12.
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 C1,γ-smooth for some γ∈(0,1). 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.  相似文献   

13.
We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ0?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.  相似文献   

14.
We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains.  相似文献   

15.
In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with d-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.  相似文献   

16.
We derive a global regularity theorem for stress fields which correspond to minimizers of convex and some special nonconvex variational problems with mixed boundary conditions on admissible domains. These are Lipschitz domains satisfying additional geometric conditions near those points, where the type of the boundary conditions changes. In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality. The regularity result for this case is derived with a difference quotient technique. In the second part the regularity results are carried over from the convex case to special nonconvex variational problems taking advantage of the relation between nonconvex variational problems and the corresponding (quasi-) convexified problems. The results are applied amongst others to the variational problems for linear elasticity, the p-Laplace operator, Hencky elasto-plasticity with linear hardening and for scalar and vectorial two-well potentials (compatible case).   相似文献   

17.
We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday  相似文献   

18.
We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincaré inequality and in addition supporting a corresponding Sobolev-Poincaré-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.  相似文献   

19.
A regularity result for solutions to boundary blow-up problems for the complex Monge–Ampère operator in balls in is proved. For certain boundary blow-up problems on bounded, strongly pseudoconvex domains in with smooth boundary an estimate of the blow-up rate of solutions are given in terms of the distance to the boundary and the product of the eigenvalues of the Levi form.  相似文献   

20.
The third problem for the Laplace equation is studied on an open set with Lipschitz boundary. The boundary condition is in Lp and it is fulfilled in the sense of the nontangential limit. The existence and the uniqueness of a solution is proved and the solution is expressed in the form of a single layer potential. For domains with C1 boundary the explicit solution of the problem is calculated.  相似文献   

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