1,720,972 research outputs found
Min-max levels on the double natural constraint
No full textA question about the possibility of using min-max methods on the double natural constraint, in spite of its lack of regularity, has been raised in some recent papers. In this note we give an answer by topological arguments which show the equivalence between constrained and unconstrained min-max classes, avoiding in this way any regularity problem
A note on compactness-type properties with respect to Lorentz norms of bounded subset of a Sobolev Space
No full textSimplified Excision Techniques for Free Discontinuity Problems in Several Variables
No full textAbstractOne of the methods proposed for the study of the problems with a free discontinuity set in two variables has been considered here under slightly different abstract assumptions from those in the previous papers. The technical difficulties are in this new setting simplified and less background is required to the reader. Moreover, applications to problems in several variables can be found thanks to a Sobolev-type theorem which ensures the Hölder continuity of certain functions out of a suitable neighborhood of their discontinuity set. Many of the results established so far for the case of two dimensions can be so extended to the general case. A new property is also proved
Concentration analysis in Banach spaces
Full text linkThe concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach-Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of Delta-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179-182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and l(p)-spaces, but not in L-p(R-N), p not equal 2 Delta-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of Delta-convergence with the Brezis-Lieb lemma and gives a version of the latter without an assumption of convergence a.
Min-Max Solutions to Some Scalar Field Equations
No full textWe show the variational structure of a multiplicity result of positive solutions u is an element of H(1) (R(N)) to the equation -Delta u + a(x)u = u(p), where N >= 2, p > 1 with p = 3 and the potential a(x) is a positive function enjoying a planar symmetry. We require suitable decay assumptions which are widely implied by those in [6], in which Wei and Yan have obtained an analogous multiplicity result by using different techniques
Concentration estimates and multiple solutions to elliptic problems at critical growth
No full textIn this paper, we consider the problem -Δu = |u|2*-2u+λu in ω, u = 0 on ∂ω, where ω is an open regular bounded subset of RN (N ≥3), 2* = 2N/N-2 is the critical Sobolev exponent and λ> 0. Our main result asserts that, if N ≥7, the problem has infinitely many solutions and, from the point of view of the compactness arguments employed here, the restriction on the dimension N cannot be weakened
Synchronic and asynchronic descriptions of irrigation problems.
No full textIn this paper we complete our work started in [31], where the present paper was announced;
in order to set a unified theory of the irrigation problem. The main result of the paper is the
equivalence of the various formulations introduced so far as well as a new one introduced
here. To this aim we introduce several geometric and analytical concepts which are essential
for reaching our final goal even if they may deserve an intrinsic interest in themselves
Elementary properties of optimal irrigation patterns
No full textIn this paper we follow the approach in Maddalena et al. (Interfaces and1
Free Boundaries 5, 391–415, 2003) to the study of the ramified structures and we
identify some geometrical properties enjoyed by optimal irrigation patterns. These
properties are “elementary” in the sense that they are not concerned with the regularity
at the ending points of such structures, where the presumable selfsimilarity
properties should take place. This preliminary study already finds an application in
G. Devillanova and S. Solimini (Math. J. Univ. Padua, to appear), where it is used in
order to discuss the irrigability of a given measure
Transport distances and irrigation models
No full textSome variational models have been recently introduced to the aim of modeling ramified structures, such as trees, rivers and so on. We introduce a general scheme in which the notion of transport distance is introduced starting from a general transport cost functional, through relaxation arguments. Then we apply this general framework to the irrigation cost, which is a particular cost functional depending on a parameter α ε]0,1[. We discuss the equivalence between this abstract approach and the above models
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