Search results for "Sobolev inequality"

Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

2011

Let $(X,d)$ be a complete, pathwise connected metric measure space with locally Ahlfors $Q$-regular measure $\mu$, where $Q>1$. Suppose that $(X,d,\mu)$ supports a (local) $(1,2)$-Poincar\'e inequality and a suitable curvature lower bound. For the Poisson equation $\Delta u=f$ on $(X,d,\mu)$, Moser-Trudinger and Sobolev inequalities are established for the gradient of $u$. The local H\"older continuity with optimal exponent of solutions is obtained.

Regularity of the inverse of a Sobolev homeomorphism in space

2006

Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

Removable sets for Sobolev spaces

1999

We study removable sets for the Sobolev space W1,p. We show that removability for sets lying in a hyperplane is essentially determined by their thickness measured in terms of a concept of p-porosity.

Maximal potentials, maximal singular integrals, and the spherical maximal function

2014

We introduce a notion of maximal potentials and we prove that they form bounded operators from L to the homogeneous Sobolev space Ẇ 1,p for all n/(n − 1) < p < n. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

Regularity of the Inverse of a Sobolev Homeomorphism

2011

We give necessary and sufficient conditions for the inverse ofa Sobolev homeomorphism to be a Sobolev homeomorphism and conditions under which the inverse is of bounded variation.

Generalized dimension estimates for images of porous sets under monotone Sobolev mappings

2014

We give an essentially sharp estimate in terms of generalized Hausdorff measures for images of porous sets under monotone Sobolev mappings, satisfying suitable Orlicz-Sobolev conditions.

Continuity of the maximal operator in Sobolev spaces

2006

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces W 1,p (R n ), 1 < p < ∞. As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

De Giorgi–Nash–Moser Theory

2015

We consider the second-order, linear, elliptic equations with divergence structure $$\mathrm{div} (\mathbb{A}(x)\nabla u(x))\;=\;\sum\limits^n_{i,j=1}\;\partial_{x_{i}}(a_{ij}(x)\partial_{x_{j}}u(x))\;=\;0.$$

Embedding of Sobolev Spaces into Lipschitz Spaces

1989

The main result of the paper is that if Ω is a bounded uniform domain in ℝn and p>n, then the Sobolev space Wl, p(Ω) embeds continously into Cα(Ω), α = 1 - n/p.

Hölder continuity of Sobolev functions and quasiconformal mappings

1993