In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^1$-valued function defined on the boundary of a bounded regular domain of $R^n$. When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology...

We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when ais radial: $a\left(\xi \right)=l\left(\right|\xi \left|\right)\left|\xi \right|\xi$ a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasingl:ℝ+ → ℝ+.

We study the problem of minimizing ${\int }_{\Omega }F\left(Du\left(x\right)\right)dx$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi$ on $\Gamma :=\partial \Omega$. The lagrangian $F$ and the domain $\Omega$ are assumed convex. A new type of hypothesis on the boundary function $\phi$ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...

In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...

We discuss the existence of solutions for a certain generalization of the membrane equation and their continuous dependence on function parameters. We apply variational methods and consider the PDE as the Euler-Lagrange equation for a certain integral functional, which is not necessarily convex and coercive. As a consequence of the duality theory we obtain variational principles for our problem and some numerical results concerning approximation of solutions.

We study the semilinear problem with the boundary reaction $$-\Delta u+u=0\text{in}\Omega ,\frac{\partial u}{\partial \nu }=\lambda f\left(u\right)\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$, $N\ge 2$, is a smooth bounded domain, $f:[0,\infty )\to (0,\infty )$ is a smooth, strictly positive, convex, increasing function which is superlinear at $\infty$, and $\lambda >0$ is a parameter. It is known that there exists an extremal parameter ${\lambda }^{*}>0$ such that a classical minimal solution exists for $\lambda <{\lambda }^{*}$, and there is no solution for $\lambda >{\lambda }^{*}$. Moreover, there is a unique weak solution $u^{*}$ corresponding to the parameter $\lambda ={\lambda }^{*}$. In this paper, we continue to study the spectral properties of $u^{*}$ and show...

An equilibrium triangular block-element, proposed by Watwood and Hartz, is subjected to an analysis and its approximability property is proved. If the solution is regular enough, a quasi-optimal error estimate follows for the dual approximation to the mixed boundary value problem of elasticity (based on Castigliano's principle). The convergence is proved even in a general case, when the solution is not regular.

We study solutions of first order partial differential relations $Du\in K$, where $u:\Omega \subset {ℝ}^n\to {ℝ}^m$ is a Lipschitz map and $K$ is a bounded set in $m\times n$ matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of $Du$ and second we replace Gromov’s $P$−convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally...