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Let $M$ be an $n$-dimensional compact Riemannian manifold without boundary and $B(r)$ a geodesic ball of radius $r$. Then for $u\in W^{1,p}(B(r))$, the Poincare and Sobolev–Poincaré inequalities are s …

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. What are the regularity results for solutions to $$ -\Delta u= \operatorname{div} F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? …

Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$. It is …

I can prove the following result. Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$. Then …

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is different …

What are the open problems in the theory of Sobolev spaces? I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to …

Is there a simple proof of the following fact? Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset W^{1-\frac{1}{n},n}(\pa …

The following paper is well known, but hard to find: J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982. In this paper …

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator $$ E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n) \quad \text{and} \quad E:W^{1,q}(\Omega)\to …