Lower semicontinuous obstacles for the porous medium equation

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.     

Kernels and Choquet capacities

Summary M. Brelot showed that the capacity corresponding to a function-kernel is a Choquet capacity, provided that the kernel satisfies the principle of equilibrium, the weak domination principle and the adjoint kernel satisfies the weak principle of equilibrium. This result is not applicable for a series of important kernels in potential theory (e.g. the fundamental solution of the heat equation, or the Kolmogorov equation), since the above principles no longer hold in this situation. New principles for function kernels guaranteeing that the capacity is a Choquet capacity are introduced and applied in the framework of balayage spaces. In particular, polar and adjoint polar sets are shown to coincide in this context.     

p-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem

For p-harmonic functions on unweighted R², with 1<p<∞, we show that if the boundary values f has a jump at an (asymptotic) corner point z₀, then the Perron solution Pf is asymptotically a+barg(z−z₀)+o(|z−z₀|). We use this to obtain a positive answer to Baernstein's problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p-harmonic measure of . We also obtain various invariance results for functions with jumps and perturbations on small sets. For p>2 these results are new also for continuous functions. Finally we look at generalizations to Rn and metric spaces.     

Positive Solutions to Singular Semilinear Elliptic Problems

We obtain a characterization of all locally bounded functions p ≥ 0 for which the equation (E) Δu +p(x)ψ(u) = 0 has a positive solution in Ω vanishing on the boundary, where Ω is a domain of ℝ^N and ψ > 0 is a nonincreasing continuous function on ]0,∞[. In particular, for Ω = ℝ^N with N ≥ 3, it is shown that (E) has a (unique) positive solution in ℝ^N which decays to zero at infinity if and only if the set {p > 0} has positive Lebesgue measure and This condition can be replaced by if p is radial.     

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.     

Fine continuity on metric spaces

We obtain pointwise estimates for solutions of obstacle problems on metric measure spaces and prove that p-superharmonic functions are p-finely continuous. Consequently, we show that p-quasicontinuous functions are p-finely continuous at p-quasievery point. As a byproduct, we obtain the sufficiency part of the Wiener criterion in metric spaces without the assumption of linear local connectedness. The author was supported by the Swedish Research Council.     

Circle boundaries of planar graphs

Letg be an infinite, connected, planar graph with bounded vertex degree, which obeys a strong isoperimetric inequality and which can be embedded in the plane so that each cycle surrounds only finitely many vertices. We investigate a certain class of compactifications ofg; one of which has boundary homemorophic to a circle. We shall show that ifg is a tree or, more generally, ifg is hyperbolic, then this circle boundary supports an integral representation of any given bounded harmonic function. We further show that in the specific case of a triangulation of the plane, the graph is hyperbolic and therefore the Martin boundary is a circle.     

Martin compactification for discrete potential theory and the mean value property

A converse of the well-known theorem on themean value property of harmonic functions is given. It is shown that a positive measurable function is harmonic if it possesses arestricted mean value property. Earlier proofs obtained using the probabilistic techniques were given by Veech, Heath and Baxter. Our approach is based on a Martin type compactification built up with the help of some quite elementarya priori inequalities foraveraging kernels.     

Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard

In this paper, the generalized Schrödinger equation (–)u=0 on the punctured unit disk of ² is investigated. If is rotation free and satisfies the Picard principle at the origin, it is shown that if a setE is minimal thin relatively to an extremal harmonic functionh with zero boundary values at {|x|=1}, there exists a sequence (r _n ) converging to zero such that B(O,r _n ) C E. Lete be the -unit. It is proved that if a measure satisfies _\E e h d<, for a minimal thin, relatively toh , setE then the Picard principle is valid for the measure + .

     

Continuous dependence on obstacles for the double obstacle problem on metric spaces

Let X be a complete metric space equipped with a doubling Borel measure supporting a p-Poincaré inequality. We obtain various convergence results for solutions of double obstacle problems on open subsets of X. In particular, we consider a sequence of double obstacle problems with converging obstacles and show that the corresponding solutions converge as well. We use the convergence properties to define and study a generalized solution of the double obstacle problem.     

Exact solution of a degenerate fully nonlinear diffusion equation

An exact solution is given for the evolution of an initially v-shaped surface by a fully nonlinear diffusion equation. This is the unique generalized solution that is continuous but not twice differentiable. Since the profile velocity decreases faster than the reciprocal of the profile curvature, the point of infinite curvature persists for a finite positive time.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.     

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.     

Bases d'ouverts réguliers et formule de Poisson pour l'équation de la chaleur

Summary Nous donnons une formule de Poisson pour certains polyèdres. Nous résolvons ainsi le problème de Dirichlet pour l'équation de la chaleur dans ces domaines.

     

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

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.     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

An Obstacle Control Problem with a Source Term

Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.     

Wiener’s criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries

This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of R^N${\mathbb{R}}^N$ (N≥3$N\ge 3$) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.     

Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form

We state a Wiener criterion for the regularity of points with respect to a relaxed Dirichlet problem for a p-homogeneous Riemannian Dirichlet form.     

An Obstacle Control Problem with a Source Term

   Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.