In this article we establish the existence of higher order weak derivatives of weak solutions of Dirichlet problem for a class of degenerate elliptic equations.

Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A\left(t\right)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.

We study a form of optimal transportation surplus functions which arise in hedonic pricing models. We derive a formula for the Ma–Trudinger–Wang curvature of these functions, yielding necessary and sufficient conditions for them to satisfy (A3w). We use this to give explicit new examples of surplus functions satisfying (A3w), of the form b(x,y) = H(x + y) where H is a convex function on ℝn. We also show that the distribution of equilibrium contracts in this hedonic pricing model is absolutely continuous...

In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of ${\mathbb{R}}^2$ is studied. Regularity properties of the solutions are proved, when the data are $L^1$-functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.

Existence of a global attractor for the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe has been shown already. In this paper we prove the higher regularity of the attractor.

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40],...

We study regularity results for solutions $u\in H{W}^{1,p}\left(\Omega \right)$ to the obstacle problem $${\int }_{\Omega }𝒜(x,{\nabla }_{ℍ}u){\nabla }_{ℍ}(v-u)\mathrm{d}x\ge 0\forall v\in {𝒦}_{\psi ,u}\left(\Omega \right)$$ such that $u\ge \psi$ a.e. in $\Omega$, where ${𝒦}_{\psi ,u}\left(\Omega \right)=\{v\in H{W}^{1,p}\left(\Omega \right):v-u\in H{W}_0^{1,p}\left(\Omega \right)v\ge \psi \text{a.e.}\text{in}\Omega \}$, in Heisenberg groups ${ℍ}^n$. In particular, we obtain weak differentiability in the $T$-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{array}{ccc}\hfill dT\psi \in H{W}_{\mathrm{loc}}^{1,p}\left(\Omega \right)& \Rightarrow Tu\in L_{\mathrm{loc}}^p\left(\Omega \right),{\left|{\nabla }_{ℍ}\psi \right|}^p\in L_{\mathrm{loc}}^q\left(\Omega \right)\hfill & \hfill \Rightarrow |{\nabla }_{ℍ}{u|}^p\in L_{\mathrm{loc}}^q\left(\Omega \right),\end{array}d$$ where $2<p<4$, $q>1$.