Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or L1L^1 data

We investigate solutions to nonlinear elliptic Dirichlet problems of the type $$\left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right.$$ where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$ and $A(x,z,\xi)$ is a Carath\'eodory's function. The growth of~the~monotone vector field $A$ with respect to the $(z,\xi)$ variables is expressed through some $N$-functions $B$ and $P$. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data and weakly monotone operator, we prove existence. For $L^1$-data problems with strongly monotone operator we infer also uniqueness and regularity of~solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces

An existence result for the Leray-Lions type operators with discontinuous coefficients

In this paper we prove an existence result for Leray-Lions quasilinear elliptic operator with discontinuous coefficients. The idea of the proof is based on compactness results for the sequences of solutions to regularized problems obtained via the Compensated Compactness, Young measures, and Set-Valued Analysis tools

The Free Material Design problem for stationary heat equation on low dimensional structures

For a given balanced distribution of heat sources and sinks, $Q$, we find an optimal conductivity tensor field, $\hat C$, minimizing the thermal compliance. We present $\hat C$ in a rather explicit form in terms of the datum. Our solution is in a cone of non-negative tensor-valued finite Borel measures. We present a series of examples with explicit solutions.49J20, %Singular parabolic equations secondary: 49K20, 80M5