1,721,022 research outputs found
Regular selections for multiple-valued functions
Full text linkGiven a multiple-valued function f, we deal with the problem of selecting its single valued branches. This problem can be stated in a rather abstract setting considering a metric space E and a finite group G of isometries of E. Given a function f which takes values in the equivalence classes of E/G, the problem consists in finding a map g with the same domain as f and taking values in E, such that at every point t the equivalence class of g(t) coincides with f(t). If the domain of f is an interval, we show the existence of a function g with these properties which, moreover, has the same modulus of continuity of f. In the particular case where E is the product of Q copies of ℝ n and G is the group of permutations of Q elements, it is possible to introduce a notion of differentiability for multiple valued functions. In this case, we prove that the function g can be constructed in such a way to preserve C k,α regularity. Some related problems are also discussed
Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions
No full textEstimates and regularity results for the DiPerna-Lions flow
Full text linkIn this paper we derive new simple estimates for ordinary differential equations
with Sobolev coefficients. These estimates not only allow to recover some old and recent
results in a simple direct way, but they also have some new interesting corollaries
Interaction of fractures in tensile bars with non-local spatial dependence
No full textWe propose to determine the displacement field u:ℐ⊂ℝ→ℝ of a 1-D bar extended in a hard device by minimizing a non-local energy functional of the type
Π[u]:=∫ ℐ Uu ' (x)+1 K∑ x i ∈J u [u](x i )ρ(x-x i )dx+∑ x i ∈J u ϕ([u](x i )),
where K is a material parameter, [u](x i ) denotes the jump of u at x i and J u ⊂ℐ is the set of all jump points. For appropriate choice of the bulk energy U(·), of the surface energy ϕ(·) and of the weight function ρ(·), we prove an existence theorem for minimizers in the space SBV(ℐ) of special bounded variation functions, and we qualitatively discuss their form by investigating the corresponding Euler-Lagrange equations. We show that, for sufficiently large values of bar elongation, minimizers of the energy are discontinuous and, most of all, the non-local term [u](x i )ρ(x-x i ) influences the relative position among the jump points, a finding that is of crucial importance to reproduce the experimental evidence
Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions
No full textA note on the Hausdorff dimension of the singular set for minimizers of the Mumford-Shah energy
No full textWe give a more elementary proof of a result by Ambrosio, Fusco and Hutchin- son to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford– Shah energy (see [1, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two authors in [4, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren’s area minimizing sets
Oscillatory solutions to transport equations
Full text linkWe show that there is no topological vector space X\subset
L^\infty\cap L^1_{\loc} (\rn{}\times \rn{n})
which embeds compactly in L^1_{\loc}, contains BV_{\loc}\cap L^\infty
and enjoys the following closure property: If f\in X^n (\rn{}\times \rn{n})
has bounded divergence and u_0\in X (\rn{n}), then there exists u\in X
(\rn{}\times \rn{n}) which solves
in the sense of distributions. X (\rn{n}) is defined as the set
of functions u_0\in L^\infty (\rn{n}) such that
belongs to X (\rn{}\times \rn{n}).
Our proof relies on an example of N.~Depauw showing an ill--posed transport equation whose vector field is ``almost ''
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