## 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.

### MSC:

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

### Keywords:

relaxation; Lavrentiev phenomenon; weighted Sobolev space
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### References:

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