An open-loop system of a multidimensional wave equation with variable coefficients, partial boundary Dirichlet control and collocated observation is considered. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The Riemannian geometry method is used in the proof of regularity and the feedthrough operator is explicitly computed.

We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution....

We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected $C$ ($\supset$-shape). Next, we show that there is a relationship between these two problems.

Para el estudio de la naturaleza de formas críticas en optimización de formas se requieren algunas propiedades de continuidad sobre las derivadas de segundo orden de las formas. Dado que la fórmula de Taylor-Young involucra a diferentes topologías que no son equivalentes, dicha fórmula no permite deducir cuando una forma crítica es un mínimo local estricto de la función forma pese a que su Hessiano sea definido positivo en ese punto. El resultado principal de este trabajo ofrece una cota superior...

In this paper we prove a regularity result for very weak solutions of equations of the type $-divA(x,u,Du)=B(x,u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x,u,Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$\left\{\begin{array}{cc}-divA(x,u,Du)=B(x,u,Du)\hfill & \text{in}\Omega ,u-u_o\in W^{1,r}(\Omega ,{ℝ}^m).\hfill \end{array}\right.$$

We study very weak solutions of an A-harmonic equation to show that they are in fact the usual solutions.

The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, $2p<n$ for a solution of a degenerate nonlinear elliptic equation in a domain $B(x_0,1)\setminus F$, $F\subset B(x_0,d)=\{x\in {ℝ}^n|x_0-x|<d\}$, $d<\frac{1}{2}$, under the boundary-value conditions $u(x,k)=k$ for $x\in \partial F$, $u(x,k)=0$ for $x\in \partial B(x_0,1)$ and where $0<\rho \le dist(x,F)$, $w\left(x\right)$ is a weighted function from some Muckenhoupt class, and $ca{p}_{p,w}\left(F\right)$, $w\left(B\right(x,\rho \left)\right)$ are weighted capacity and measure of the corresponding sets.