Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type

In this paper, the Hölder regularity of weak solutions for singular parabolic systems of p-Laplacian type is investigated. By the Poincare inequality, we show that its weak solutions within Hölder space.

Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

The trichotomy between regular, semiregular, and strongly irregular boundary points for $$p$$ p -harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $$p$$ p -Poincaré inequality, $$1<p<\infty $$ 1 < p < ∞ . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $$p$$ p -harmonic measures, removability, and semibarriers.

BV and Sobolev homeomorphisms between metric measure spaces and the plane

We show that, given a homeomorphism f : G → Ω {f:G\rightarrow\Omega} where G is an open subset of ℝ 2 {\mathbb{R}^{2}} and Ω is an open subset of a 2-Ahlfors regular metric measure space supporting a weak ( 1 , 1 ) {(1,1)} -Poincaré inequality, it holds f ∈ BV loc ⁡ ( G , Ω ) {f\in{\operatorname{BV_{\mathrm{loc}}}}(G,\Omega)} if and only if f - 1 ∈ BV loc ⁡ ( Ω , G ) {f^{-1}\in{\operatorname{BV_{\mathrm{loc}}}}(\Omega,G)} . Further, if f satisfies the Luzin N and N - 1 {{}^{-1}} conditions, then f ∈ W loc 1 , 1 ⁡ ( G , Ω ) {f\in\operatorname{W_{\mathrm{loc}}^{1,1}}(G,\Omega)} if and only if f - 1 ∈ W loc 1 , 1 ⁡ ( Ω , G ) {f^{-1}\in\operatorname{W_{\mathrm{loc}}^{1,1}}(\Omega,G)} .

Functional Inequalities for Two-Level Concentration

Probability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary

We establish several useful estimates for a non-conforming 2-norm posed on piecewise linear surface triangulations with boundary, with the main result being a Poincaré inequality. We also obtain equivalence of the non-conforming 2-norm posed on the true surface with the norm posed on a piecewise linear approximation. Moreover, we allow for free boundary conditions. The true surface is assumed to be C 2 , 1 C^{2,1} when free conditions are present; otherwise, C 2 C^{2} is sufficient. The framework uses tools from differential geometry and the closest point map (see [G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155]) for approximating the full surface Hessian operator. We also present a novel way of applying the closest point map when dealing with surfaces with boundary. Connections with surface finite element methods for fourth-order problems are also noted.

Generalization of Poincar ´e inequality in a Sobolev Space with exponent constant to the case of Sobolev space with a variable exponent

In this work, we study the Poincare inequality in Sobolev spaces with variable exponent. As a consequence of this ´ result we show the equivalent norms over such cones. The approach we adopt in this work avoids the difficulty arising from the possible lack of density of the space C∞ 0 (Ω).

Poincaré Inequality on Subanalytic Sets

Let $$\Omega $$ Ω be a subanalytic connected bounded open subset of $$\mathbb {R}^n$$ R n , with possibly singular boundary. We show that given $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) , there is a constant C such that for any $$u\in W^{1,p}(\Omega )$$ u ∈ W 1 , p ( Ω ) we have $$||u-u_{\Omega }||_{L^p} \le C||\nabla u||_{L^p},$$ | | u - u Ω | | L p ≤ C | | ∇ u | | L p , where we have set $$u_{\Omega }:=\frac{1}{|\Omega |}\int _{\Omega } u.$$ u Ω : = 1 | Ω | ∫ Ω u .