The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of the reference...

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed...

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri...

Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function $f$ defined on a separable Banach space are studied. The conditions are in terms of a majorization of $f$ by a $C^1$-smooth function, separability of the boundary for $f$ or an approximation of $f$ by Fréchet smooth convex functions.

We study the Dirichlet boundary value problem for the $p$-Laplacian of the form $$-{\Delta }_{p}u-{\lambda }_1{\left|u\right|}^{p-2}u=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $N\ge 1$, $p>1$, $f\in C\left(\overline{\Omega }\right)$ and ${\lambda }_1>0$ is the first eigenvalue of ${\Delta }_p$. We study the geometry of the energy functional $$E_p\left(u\right)=\frac{1}{p}{\int }_{\Omega }{\left|\nabla u\right|}^p-\frac{{\lambda }_1}{p}{\int }_{\Omega }{\left|u\right|}^p-{\int }_{\Omega }fu$$ and show the difference between the case $1<p<2$ and the case $p>2$. We also give the characterization of the right hand sides $f$ for which the above Dirichlet problem is solvable and has multiple solutions.

In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set ${𝒰}_{\mathrm{a}d}$ of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by ${𝒰}_{\mathrm{a}d}$ and maximizes a functional criterion reflecting a particular feature of the state solution, as local stress, displacement, or temperature, for instance. An increase in the criterion value indicates a deterioration in the featured quantity....

The present part of the paper completes the discussion in Part I in two directions. Firstly, in Section 5 a number of existence theorems for a solution to Problem III (principle of minimum potential energy) is established. Secondly, Section 6 and 7 are devoted to a discussion of both the classical and the abstract approach to the duality theory as well as the relationship between the solvability of Problem III and its dual one.

We characterize some $G$-limits using two-scale techniques and investigate a method to detect deviations from the arithmetic mean in the obtained $G$-limit provided no periodicity assumptions are involved. We also prove some results on the properties of generalized two-scale convergence.

We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality. As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.

The paper deals with the problem of finding a curve, going through the interior of the domain $\Omega$, accross which the flux $\partial u/\partial n$, where $u$ is the solution of a mixed elliptic boundary value problem solved in $\Omega$, attains its maximum.