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Relaxation of degenerate variational integrals. (English) Zbl 0799.49012

The lower semicontinuous envelope \(\overline F\) of the functional \[ F(u)=\begin{cases} \int_ \Omega f(x,Du)dx & \text{if } u\in {\mathcal C}^ 1(\Omega),\\+\infty &\text{if } u\in W^{1,1}(\Omega)\backslash{\mathcal C}^ 1(\Omega)\end{cases} \] is computed when \(\lambda(x)| \xi|^ p\leq f(x,\xi)\leq \Lambda(x)(1+ | \xi|^ p)\) for a.e. \(x\in \mathbb{R}^ n\), \(\forall\xi\in \mathbb{R}^ n\), \({\Lambda\over \lambda}\in L^ \infty(\mathbb{R}^ n)\) and \(\lambda\) is a weight in the Muckenhoupt class \(A_ p\). In this case, \(\overline F(u)= \int_ \Omega f(x,Du)dx\) for every \(u\in W^{1,1}(\Omega)\), and the related minimum problems are not affected by the Lavrentiev phenomenon. The proof is based on approximation by convolution in a suitable weighted Sobolev space. A weight \(\lambda\) that is not in the \(A_ p\)-class, for which the approximation by convolution does not converge, is also explicitly constructed.
Reviewer: P.Chiadò

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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