Search results for "Potential theory"
showing 10 items of 24 documents
Failure of topological rigidity results for the measure contraction property
2014
We give two examples of metric measure spaces satisfying the measure contraction property MCP(K,N) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0,3) and contains a subset isometric to $\mathbb{R}$, but does not topologically split. The second space satisfies MCP(2,3) and has diameter $\pi$, which is the maximal possible diameter for a space satisfying MCP(N-1,N), but is not a topological spherical suspension. The latter example gives an answer to a question by Ohta.
Superharmonic functions are locally renormalized solutions
2011
Abstract We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.
Equivalence of viscosity and weak solutions for the $p(x)$-Laplacian
2010
We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem.
Quasi-Continuous Vector Fields on RCD Spaces
2021
In the existing language for tensor calculus on RCD spaces, tensor fields are only defined $\mathfrak {m}$ -a.e.. In this paper we introduce the concept of tensor field defined ‘2-capacity-a.e.’ and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.
H�lder continuity of solutions to quasilinear elliptic equations involving measures
1994
We show that the solutionu of the equation $$ - div(|\nabla u|^{p - 2} \nabla u) = \mu $$ is locally β-Holder continuous provided that the measure μ satisfies the condition μ(B(x,r))⩽Mrn − p + α(p − 1) for some α>β. A corresponding result for more general operators is also proven.
The Obstacle Problem in a Non-Linear Potential Theory
1988
M. Brelot gave rise to the concept harmonic space when he extended classical potential theory on ℝn to an axiomatic system on a locally compact space. I have recently constructed1 a non-linear harmonic space by dropping the assumption that the sum of two harmonic functions is harmonic and considering some other axioms instead. This approach has its origin in the work of O. Martio, P. Lindqvist and S. Granlund2,3,4, who have developed a non-linear potential theory on ℝn connected with variational integrals of the type ∫ F(x,∇u(x)) dm(x), where F(x, h) ≈ |h|p.
Polar Sets in a Nonlinear Potential Theory
1988
In this lecture we discuss nonlinear potential theory based on “A-super-harmonic functions”; the theory can be viewed as a (nonlinear) extension of the classical study of superharmonic functions in ℝn.
Spontaneous periodic and bursting current oscillations in iron corrosion by dichromate: a useful study for simulating biological systems
1995
Abstract Studies on chemical and electrochemical oscillating systems are very useful in understanding more complex biological systems. Spontaneous periodic and bursting current oscillations were found in iron/dichromate systems coupled with graphite or zinc electrodes. In this paper, we study some phenomenological features of the two systems: their typical oscillation profiles and the influence of external resistance. The results are explained by the Franck-Fitzhugh model using the mixed potential theory.
Singular Neumann (p, q)-equations
2019
We consider a nonlinear parametric Neumann problem driven by the sum of a p-Laplacian and of a q-Laplacian and exhibiting in the reaction the competing effects of a singular term and of a resonant term. Using variational methods together with suitable truncation and comparison techniques, we show that for small values of the parameter the problem has at least two positive smooth solutions.
A note on an overdetermined problem for the capacitary potential
2016
We consider an overdetermined problem arising in potential theory for the capacitary potential and we prove a radial symmetry result.