We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the...

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t+{\nabla }_g·f(x,u)=0$ on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}}$ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$

We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result...

We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space ${\mathbb{R}}_{+}^2$ of ${\mathbb{R}}^2$. We show that for a particular initial datum, which is Lipschitz continuous and bounded on ${\mathbb{R}}_{+}^2$, the second derivative of the classical solution is not uniformly continuous on ${\mathbb{R}}_{+}^2$. In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.

We construct a bounded domain $\Omega \subset {ℝ}^2$ with the cone property and a harmonic function on Ω which belongs to $W_0^{1,p}\left(\Omega \right)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no $L^p$-Hodge decomposition in $L^p(\Omega ,{ℝ}^2)$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in $W^{1,p}\left(\Omega \right)$ for all p > 4.

We modify an example due to X.-J. Wang and obtain some counterexamples to the regularity of the degenerate complex Monge-Ampère equation on a ball in ℂⁿ and on the projective space ℙⁿ.

We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for...

The celebrated criterion of Petrowsky for the regularity of the latest boundary point, originally formulated for the heat equation, is extended to the so-called p-parabolic equation. A barrier is constructed by the aid of the Barenblatt solution.

We consider the equation $u_t\mathrm{-}i\mathrm{\Lambda }u=0$, where $\mathrm{\Lambda }=\lambda \left(D_x\right)$ is a first order pseudo-differential operator with real symbol $\lambda \left(\xi \right)$. Under a suitable convexity assumption on $\lambda$ we find the decay properties for $u\left(t,x\right)$. These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem $u_t\mathrm{-}i\mathrm{\Lambda }u=f\left(u\right)$, $u\left(0,x\right)=g\left(x\right)$. If $f\left(u\right)$ is a smooth function which satisfies $f\left(u\right)\simeq {\left|u\right|}^p$ near $u=0$, and $g$ is small in suitably Sobolev norm, we prove global existence theorems provided $p$ is greater than a critical exponent.