Hölder estimates of p-harmonic extension operators

It is now a well-known fact that for 1<p<∞ the p-harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a (1,p)-Poincaré inequality are locally Hölder continuous. In this note we provide a characterization of domains in such metric spaces for which p-harmonic extensions of Hölder continuous boundary data are globally Hölder continuous. We also provide a link between this regularity property of the domain and the uniform p-fatness of the complement of the domain.     

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.     

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?.     

Partial regularity for degenerate subelliptic systems associated with Hörmander’s vector fields

The paper is devoted to partial Hölder estimates for weak solutions of a class of degenerate subelliptic systems constructed by Hörmander’s vector fields. Assumptions on the systems are that coefficient matrix satisfies VMO condition in the sense of Carnot-Carathéodory distance in variables x for fixed u, and lower order terms satisfy the natural growth condition and the controllable growth condition, respectively.     

Existence and uniqueness analysis of a detached shock problem for the potential flow

We study a problem for two-dimensional steady potential and isentropic Euler equations in a bounded domain, where an artificial detached shock interacts with a wedge. Using the stream function, we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted Hölder spaces. The proof is based on various Hölder and Schauder estimates for second-order elliptic equations and fixed point theorems. Moreover, we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional.     

Asymptotically regular problems I: Higher integrability

We consider weak solutions u of non-linear systems of partial differential equations. Assuming that the system exhibits a certain kind of elliptic behavior near infinity we prove higher integrability results for the gradient Du. In particular, we establish Hölder continuity of u in low dimensions. Moreover, we obtain analogous results for vectorial minimizers of multi-dimensional variational integrals. Finally, we discuss an extension to minimizing sequences and applications to generalized minimizers.     

The transmission problem in domains with a corner point for the Laplace operator in weighted Hölder spaces

We prove the existence and uniqueness of a solution to the elliptic transmission problem in nonsmooth domains in the weighted Hölder space. The coercive estimates of the solution are given.     

Hölder continuity of weak solutions of a class of linear equations with boundary degeneracy

This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison principles, we investigate the optimal Hölder continuity of weak solutions in these two cases utilizing the Harnack inequality and the Morrey theorem, respectively.     

The Muskat problem with surface tension and a nonregular initial interface

In this paper, we consider the two-dimensional Muskat problem with surface tension on a free boundary. The initial shape of the unknown interface has a corner point. We prove that the problem has a unique solution in the weighted Hölder classes locally in time and specify the sufficient conditions for the existence of the “waiting time” phenomenon.     

Oblique boundary value problems for equations of Monge-Ampère type

We prove the existence of classical solutions of elliptic equations of Monge-Ampère type subject to a semilinear oblique boundary condition which is a perturbation of the Neumann boundary condition. Our techniques also allow us to treat fully nonlinear strictly oblique boundary conditions satisfying a concavity condition. Examples show that the above restrictions on the boundary condition are generally necessary for the existence of classical solutions. Received May 22, 1996 / Accepted April 10, 1997     

Existence of solutions to quasi-linear elliptic problems with generalized Robin boundary conditions

We characterize the L^q-solvability of a class of quasi-linear elliptic equations involving the p-Laplace operator with generalized nonlinear Robin type boundary conditions on bad domains. Some uniqueness results are also given.     

Reverse Hölder inequalities for singular parabolic equations near the boundary

We show that weak solutions to a singular parabolic partial differential equation globally belong to a higher Sobolev space than assumed a priori. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. The results extend to singular parabolic systems as well. Motivation for studying reverse Hölder inequalities comes partly from applications to regularity theory.     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Harnack estimates for a quasi-linear parabolic equation with a singular weight

In this paper, based on measure theoretical arguments, we establish Harnack estimates and Hölder continuity of nonnegative weak solutions for a degenerate parabolic equation with a singular weight. We transform the equation by performing the change of function. The energy estimates, the upper boundedness, the lower boundedness and the expansion of positivity for the solutions to the transformed equation are obtained. Then our aim is reached.     

The Calderón-Zygmund property for quasilinear divergence form equations over Reifenberg flat domains

The results by Palagachev (2009) [3] regarding global Hölder continuity for the weak solutions to quasilinear divergence form elliptic equations are generalized to the case of nonlinear terms with optimal growths with respect to the unknown function and its gradient. Moreover, the principal coefficients are discontinuous with discontinuity measured in terms of small BMO norms and the underlying domain is supposed to have fractal boundary satisfying a condition of Reifenberg flatness. The results are extended to the case of parabolic operators as well.     

The regularity for nonlinear parabolic systems of p-Laplacian type with critical growth

We study the Hölder regularity of weak solutions to the evolutionary p  -Laplacian system with critical growth on the gradient. We establish a natural criterion for proving that a small solution and its gradient are locally Hölder continuous almost everywhere. Actually our regularity result recovers the classical result in the case p=2$p=2$ [16] and can be applied to study the regularity of the heat flow for m-dimensional H-systems as well as the m-harmonic flow.     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

A survey of results on nonlinear Venttsel problems

We review the recent results for boundary value problems with boundary conditions given by second-order integral-differential operators. Particular attention has been paid to nonlinear problems (without integral terms in the boundary conditions) for elliptic and parabolic equations. For these problems we formulate some statements concerning a priori estimates and the existence theorems in Sobolev and Hölder spaces.