# Formation of cracks under deformations with .nite energy

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In this paper we deal with the attainment and relaxation issues in variational problems of mathematical theory of elasticity.We consider minimization of energy functionals in certain classes of deformations which render the problem essentially scalar. It turns out that in those cases the...

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Estimates the Hausdorff dimension of the singular set of solutions to non-differentiable elliptic systems. Elliptic system in divergence form; Bridge built between the Hï¿½lder and Lipschitz continuity; Properties of fractional Sobolev space.

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Suppose that ( X, ï¿½, d) is a space of homogeneous type, where d is themetric and ï¿½ is the measure related by the doubling condition with exponent ? > 0, W( X), p > 1, are the generalized Sobolev classes, a > 0, and dimH is the Hausdorff dimension. We prove that, for any function u ? W(...

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Here the authors are interested in the zero set of Sobolev functions and functions of bounded variation with negative power of integrability. The main result is a general Hausdorff dimension estimate on the size of zero set. The research is motivated by the model on van der waal force driven...

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In this paper, we present several extensions of theoretical tools for the analysis of discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1, 8), and we prove generalizations of many techniques of classical analysis in Sobolev spaces....

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In this note we take the view that compactness in Lp can be seen quantitatively on a scale of fractional Sobolev type spaces. To accommodate this viewpoint one must work on a scale of spaces, where the degree of differentiability is measured, not by a power function, but by an arbitrary function...

- Local Boundedness of Minimizers with Limit Growth Conditions. Cupini, Giovanni; Marcellini, Paolo; Mascolo, Elvira // Journal of Optimization Theory & Applications;Jul2015, Vol. 166 Issue 1, p1
The energy integral of the calculus of variations, which we consider in this paper, has a limit behavior when the maximum exponent $$q$$ , in the growth estimate of the integrand, reaches a threshold. In fact, if $$q$$ is larger than this threshold, counterexamples to the local boundedness and...