Local Hölder continuity of weak solutions for an anisotropic elliptic equation

We prove, following DiBenedetto’s intrinsic scaling method, that a local bounded weak solution of the equation $$-\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\left[\left| \frac{\partial u}{\partial x_i} \right|^{p_{i}-2} \frac{\partial u}{\partial x_i} \right]=f $$ in Ω, is locally Hölder continuous, where f is a given bounded function and p _i  ≥ 2, for any i = 1, . . . , N.     

Local Hölder continuity of nonnegative weak solutions of degenerate parabolic equations

In this paper, we consider nonnegative weak solutions for a class of degenerate parabolic equations. Using Moser’s method, we get the local boundedness of solutions to equations of this class. Then we prove that the solutions are locally Hölder continuous.     

Hölder continuity of solutions of an elliptic equation uniformly degenerating on part of the domain

We consider a second-order divergence elliptic equation in a domain D divided by a hyperplane in two parts. The equation is uniformly elliptic in one of these parts and is uniformly degenerate with respect to a small parameter ? in the other. We show that each solution is Hölder continuous in D with Hölder exponent independent of ?.     

Hölder continuity of solutions to quasilinear elliptic equations involving measures

We show that the solutionu of the equation

Hölder continuity of solutions to a degenerate elliptic equation in the presence of the Lavrentiev phenomenon

We study a second-order elliptic equation for which the Dirichlet problem can be posed in a nonunique way due to the so-called Lavrentiev phenomenon. In the corresponding weighted Sobolev space smooth functions are not dense, which leads to the existence of W – solutions and H – solutions. For H - solutions, we establish the Hölder continuity. We also discuss this question for W – solutions, for which the situation is more complicated.     

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.     

On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems

Summary The Hölder continuity of bounded weak solutions of quasilinear parabolic systems with main part in diagonal form is proved via a parabolic hole-filling technique.This research was supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft     

Hölder continuity of solutions to elastic traffic network models

This paper aims to study stability and sensitivity analysis for quasi-variational inequalities which model traffic network equilibrium problems with elastic travel demand. In particular, we provide a Hölder stability result under parametric perturbations.     

The Hölder continuity of the solution map to the b-family equation in weak topology

We prove that the solution map of the $b$ -family equation is Hölder continuous as a map from a bounded set of $H^s(\mathbb{R }), s>\frac{3}{2}$ with $H^r(\mathbb{R })$ ( $0\le r<s$ ) topology, to $C([0, T], H^r(\mathbb{R }))$ for some $T>0$ . Moreover, we show that the obtained exponent of the Hölder continuity is optimal when $s-1<r<s$ .     

Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set

In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem

$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$

where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).

     

Hölder continuity of interfaces for the porous medium equation with absorption

In this paper, we study the one-dimensional motion of viscous gas connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coefficient μ is proportional to ρ^θ and 0 < θ < 1/2, where ρ is the density, the global existence and the uniqueness of weak solutions are proved. This improves the previous results by enlarging the interval of θ.     

Uniform Hölder continuity of approximate solutions to parabolic systems and its application

By means of a minimizing scheme, we constructively define approximate solutions to parabolic systems. On the basis of Campanato theory, these approximate solutions are shown to be locally Hölder equi-continuous with respect to an approximation parameter. As an application, we obtain a weak solution to an initial-boundary value problem with the same Hölder continuity.     

Hölder continuity of solutions to a basic problem in the calculus of variations

For the basic problem in the calculus of variations where the Lagrangian is convex and depends only on the gradient, we establish the continuity of the solutions when the Dirichlet boundary condition is defined by a continuous function ϕ. When ϕ is Lipschitz continuous, then the solutions are Hölder continuous. To cite this article: P. Bousquet et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions

Hölder continuity and uniqueness of the solutions of general multivalued vector quasiequilibrium problems in metric spaces are established. The results are shown to be extensions of recent ones for equilibrium problems with some improvements. Applications in quasivariational inequalities, vector quasioptimization and traffic network problems are provided as examples for others in various optimization—related problems.     

On Hölder continuity of solutions for a class of nonlinear elliptic systems with $$p$$ p -growth via weighted integral techniques

Annali di Matematica Pura ed Applicata (1923 -) - We consider weak solutions of nonlinear elliptic systems in a $$W^{1,p}$$ -setting which arise as Euler–Lagrange equations of certain...     

Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent

In this paper, we prove the Hölder continuity of solutions of parabolic equations containing the p(x, t)-Laplacian. The degree p must satisfy the so-called logarithmic condition.