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Minimizers of variational problems with Euler-Lagrange equations having measure-valued right-hand side. (English) Zbl 0965.49013

Summary: We define minimizers of the problem \[ J(u):= \int_\Omega L(x,Du(x)) dx- \int_\Omega u(x) d\mu\to \min,\;u|_{\partial\Omega}= 0, \] where \(L(x,v): \Omega\times \mathbb{R}^n\to \mathbb{R}\) satisfies standard growth conditions and is convex in \(v, \Omega\subset\mathbb{R}^n\), and \(\mu\) is a Radon measure with finite mass on \(\Omega\), as limit points \((M\to \infty)\) of solutions \(u_M\) to the problems \[ J(u)\to \min,\;u\in W^{1,\infty}_0(\Omega),\;\|Du\|_{L^\infty(\Omega; \mathbb{R}^m)}\leq M. \] We prove that minimizers are distributional solutions of the problem \[ -\text{div }L_v(x, Du)= \mu\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] with some fine properties. In particular, a minimizer is unique and is an entropy solution if \(\mu\) does not charge sets of zero \(p\)-capacity.
The result follows from properties of the cost-value function \(I(M):= J(u_M)\), the derivative of which controls the behavior of the above differential operator at \(u_M\), and some technical auxiliary constructions with Sobolev functions.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35J25 Boundary value problems for second-order elliptic equations
49J20 Existence theories for optimal control problems involving partial differential equations
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