1,720,973 research outputs found
Γ-convergence for power-law functionals with variable exponents
No full textWe study the Γ-convergence of the functionals Fn(u):=||f(⋅,u(⋅),Du(⋅))||pjavax.xml.bind.JAXBElement@1599978c(⋅) and [Formula presented] defined on X∈{L1(Ω,Rd),L∞(Ω,Rd),C(Ω,Rd)} (endowed with their usual norms) with effective domain the Sobolev space W1,pjavax.xml.bind.JAXBElement@17c9c5d8(⋅)(Ω,Rd). Here Ω⊆RN is a bounded open set, N,d≥1 and the measurable functions pn:Ω→[1,+∞) satisfy the conditions ess supΩpn≤βess infΩpn<+∞ for a fixed constant β>1 and ess infΩpn→+∞ as n→+∞. We show that when f(x,u,⋅) is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as n→∞, the sequence (Fn)nΓ-converges in X to the functional F represented as F(u)=||f(⋅,u(⋅),Du(⋅))||∞ on the effective domain W1,∞(Ω,Rd). Moreover we show that the Γ-limnFn is given by the functional F(u):=0if||f(⋅,u(⋅),Du(⋅))||∞≤1,+∞otherwiseinX
A Relaxation Result in the Vectorial Setting and Power Law Approximation for Supremal Functionals
No full textWe provide relaxation for not lower semicontinuous supremal functionals defined on vectorial Lipschitz functions, where the Borel level convex density depends only on the gradient. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally, we discuss the power law approximation of supremal functionals, with nonnegative, coercive densities having explicit dependence also on the spatial variable, and satisfying minimal measurability assumptions
Standing waves for a class of Schr"odinger equations with potentials in
No full textWe prove the existence of standing waves to the following family of nonlinear Schrödinger equations: [ ih∂_tψ = −h2Δψ + V (x)ψ − ψ|ψ|^{p−2}, (t, x) ∈ R × R^nh > 02 < p < 2n/(n − 2)n ≥ 32 < p < ∞n = 1, 2V (x) ∈ L^∞(R^n)$ is assumed to have a sublevel with positive and finite measure
On Morrey's inequality in Sobolev-Slobodeckiĭ spaces
No full textWe study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodeckiĭ spaces on the whole RN. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some regularity estimates. We also analyze the sharp asymptotic behaviour of this constant as we reach the borderline case sp=N, where the inequality fails. This can be done by means of a new elementary proof of the Morrey inequality, which combines: a local fractional Poincaré inequality for punctured balls, the definition of capacity of a point and Hardy's inequality for the punctured space. Finally, we compute the limit of the sharp Morrey constant for s↗1, as well as its limit for p↗∞. We obtain convergence of extremals, as well
On a class of Cheeger inequalities
No full textWe study a general version of the Cheeger inequality by considering the shape functional F-p,F-q(Omega) = lambda(1/p)(p)(Omega)/lambda q(1/q)(Omega). The infimum and the supremum of F-p,F-q are studied in the class of all domains Omega of R-d and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for F-p,F-q is discussed
Sobolev embeddings and distance functions
No full textOn a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 1, p 0 into L q and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when p is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when p is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the p-Laplacian with sub-homogeneous right-hand side, as the exponent p diverges to ∞. The case of first eigenfunctions of the p-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies
A comparison principle for the Lane–Emden equation and applications to geometric estimates
No full textWe prove a comparison principle for positive supersolutions and subsolutions to the Lane–Emden equation for the p-Laplacian, with subhomogeneous power in the right-hand side. The proof uses variational tools and the result applies with no regularity assumptions, both on the set and the functions. We then show that such a comparison principle can be applied to prove: uniqueness of solutions; sharp pointwise estimates for positive solutions in convex sets; localization estimates for maximum points and sharp geometric estimates for generalized principal frequencies in convex sets
Some Inequalities Involving Perimeter and Torsional Rigidity
No full textWe consider shape functionals of the form Fq(Ω) = P(Ω) Tq(Ω) on the class of open sets of prescribed Lebesgue measure. Here q> 0 is fixed, P(Ω) denotes the perimeter of Ω and T(Ω) is the torsional rigidity of Ω. The minimization and maximization of Fq(Ω) is considered on various classes of admissible domains Ω : in the class Aall of all domains, in the class Aconvex of convex domains, and in the class Athin of thin domains
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