We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type $$𝒥\left(v\right)={\int }_{\Omega }j(x,v,\nabla v)\mathrm{d}x+{\int }_{\Omega }a\left(x\right){\left|v\right|}^2\mathrm{d}x-{\int }_{\Omega }fv\mathrm{d}x,v\in W_0^{1,2}\left(\Omega \right),$$ where $\Omega \subset {ℝ}^N$, $j$ is a Carathéodory function such that $\xi \mapsto j(x,s,\xi )$ is convex, and there exist constants $0\le \tau <1$ and $M>0$ such that $$\frac{{\left|\xi \right|}^2}{{(1+|s\left|\right)}^{\tau }}\le j(x,s,\xi )\le M{\left|\xi \right|}^2$$ for almost all $x\in \Omega$, all $s\in ℝ$ and all $\xi \in {ℝ}^N$. We show that, even if $0<a\left(x\right)$ and $f\left(x\right)$ only belong to $L^1\left(\Omega \right)$, the interplay $$\left|f\right(x\left)\right|\le 2Qa\left(x\right)$$ implies the existence of a minimizer $u\in W_0^{1,2}\left(\Omega \right)$ which belongs to $L^{\infty }\left(\Omega \right)$.

Sea f: N → R una función convexa y sea x ∈ Ni, donde N es un convexo en un espacio vectorial real. Se demuestra que, si Df&lt;(x) es no vacío, entonces Df&lt;(x) es el interior algebraico de Df≤(x).

In this paper, we introduce a new class of variational inequality with its weak and split forms to obtain an $LU$-optimal solution to the multi-dimensional interval-valued variational problem, which is a wider class of interval-valued programming problem in operations research. Using the concept of (strict) $LU$-convexity over the involved interval-valued functionals, we establish equivalence relationships between the solutions of variational inequalities and the (strong) $LU$-optimal solutions of the multi-dimensional...

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean...

We study properties of the functional$$\begin{array}{c}\hfill {ℱ}_{\mathrm{loc}}(u,\Omega ):=\underset{\left(u_j\right)}{inf}\left\{\underset{j\to \infty }{lim\; inf}{\int }_{\Omega }f\left(\nabla u_j\right)x\left|\begin{array}{cc}& \left(u_j\right)\subset W_{\mathrm{loc}}^{1,r}\left(\Omega ,\right)\hfill \\ & u_{j}u\mathrm{in}\left(\Omega ,\right)\hfill \end{array}\right.\right\},\end{array}$$F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;RN), and f:RN × n → R is continuous and satisfies 0 ≤ f(ξ) ≤ L(1 + | ξ | r). For r ∈ [1,2), assuming f has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc. Then, for $r\in [1,\frac{n}{n-1})$ r ∈ [ 1 , n n − 1 ) , we prove that...

In the framework of the linear fracture theory, a commonly-used tool to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called energy release rate defined as the variation of the mechanical energy with respect to the crack dimension. Precisely, the well-known Griffith's criterion postulates the evolution of the crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we consider the shape design problem which consists in optimizing...

As a model for the energy of a brittle elastic body we consider an integral functional consisting of two parts: a volume one (the usual linearly elastic energy) which is quadratic in the strain, and a surface part, which is concentrated along the fractures (i.e. on the discontinuities of the displacement function) and whose density depends on the jump part of the strain. We study the problem of the lower semicontinuous envelope of such a functional under the assumptions that the surface energy density...

Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and $\varphi \in L^1(\partial \Omega ,{\mathscr{H}}^{n-1})$, we prove an explicit representation formula for the L1 lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint $u^{+}\ge \psi$${\mathscr{H}}^{...}$

In this paper we study the lower semicontinuous envelope with respect to the $L^1$-topology of a class of isotropic functionals with linear growth defined on mappings from the $n$-dimensional ball into ${ℝ}^N$ that are constrained to take values into a smooth submanifold $𝒴$ of ${ℝ}^N$.

In this paper we study the lower semicontinuous envelope with respect to the L1-topology of a class of isotropic functionals with linear growth defined on mappings from the n-dimensional ball into  ${ℝ}^N$  that are constrained to take values into a smooth submanifold  $𝒴$  of  ${ℝ}^N$.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an $L^p$-space ($p\&lt;\infty$). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider the weak closure $WZ$ of the set $Z$ of all feasible pairs (solution, flow) of the family of potential elliptic systems$$$$where $\Omega \subset {𝐑}^n$ is a bounded Lipschitz domain, $F_s$ are strictly convex smooth functions with quadratic growth and $S=\{\sigma measurable\mid {\sigma }_s\left(x\right)=0\text{or}1,s=1,\cdots ,s_0,{\sigma }_1\left(x\right)+\cdots +{\sigma }_{s_0}\left(x\right)=1\}$. We show that $WZ$ is the zero level set for an integral functional with the integrand $Qℱ$ being the $𝐀$-quasiconvex envelope for a certain function $ℱ$ and the operator $𝐀={\left(\text{curl,div}\right)}^m$. If the functions $F_s$ are isotropic, then on the characteristic cone $\Lambda$ (defined by the operator $𝐀$) $Qℱ$ coincides...