1,721,024 research outputs found
A symmetry result in R2 for global minimizers of a general type of nonlocal energy
Full text linkIn this paper, we are interested in a general type of nonlocal energy, defined on a ball BR⊂ Rn for some R> 0 as E(u,BR)=∫∫R2n\(CBR)2F(u(x)-u(y),x-y)dxdy+∫BRW(u)dx.We prove that in R2, under suitable assumptions on the functions F and W, bounded continuous global energy minimizers are one-dimensional. This proves a De Giorgi conjecture for minimizers in dimension two, for a general type of nonlocal energy
Some observations on the Green function for the ball in the fractional Laplace framework
Full text linkWe consider a fractional Laplace equation and we give a selfcontained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new, however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree cour
The stickiness phenomena of nonlocal minimal surfaces: new results and a comparison with the classical case
Full text linkWe discuss in this note the stickiness phenomena for nonlocal minimal surfaces. Classical minimal surfaces in convex domains do not stick to the boundary of the domain, hence examples of stickiness can be obtained only by removing the assumption of convexity. On the other hand, in the nonlocal framework, stickiness is ''generic''. We provide various examples from the literature, and focus on the case of complete stickiness in highly nonlocal regimes
Local density of Caputo-stationary functions in the space of smooth functions
Full text linkWe consider the Caputo fractional derivative and say that a function is Caputo-stationary if its Caputo derivative is zero. We then prove that any Ck( [ 0,1 ]) function can be approximated in [0,1] by a function that is Caputo-stationary in [0,1], with initial point a< 0. Otherwise said, Caputo-stationary functions are dense in Ckloc(R)
A fractional elliptic problem in R n with critical growth and convex nonlinearities
No full textIn this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in R n (-Δ)su=εhuq+u2s∗-1in the convex case 1≤q<2s∗-1, where 2s∗=2n/(n-2s) is the critical fractional Sobolev exponent, (- Δ) s is the fractional Laplace operator, ε is a small parameter and h is a given bounded, integrable function. The problem has a variational structure and we prove the existence of a solution by using the classical Mountain-Pass Theorem. We work here with the harmonic extension of the fractional Laplacian, which allows us to deal with a weighted (but possibly degenerate) local operator, rather than with a nonlocal energy. In order to overcome the loss of compactness induced by the critical power we use a Concentration-Compactness principle. Moreover, a finer analysis of the geometry of the energy functional is needed in this convex case with respect to the concave–convex case studied in Dipierro et al. (Fractional elliptic problems with critical growth in the whole of R n . Lecture Notes Scuola Normale Superiore di Pisa, vol 15. Springer, Berlin, 2017)
On the mean value property of fractional harmonic functions
Full text linkAs is well known, harmonic functions satisfy the mean value property, i.e. the average of such a function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a feature characterizing only balls, namely, is a set, for which all harmonic functions satisfy the mean value property, necessarily a ball? This question was investigated by several authors, including Bernard Epstein (1962), Bernard Epstein and Schiffer (1965), Myron Goldstein and Wellington (1971), who obtained a positive answer to this question under suitable additional assumptions. The problem was finally elegantly, completely and positively settled by Ülkü Kuran (1972), with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, et al. (in press) who proved a quantitative stability result for the mean value formula, showing that a suitable “mean value gap” (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the “gap” between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain “almost” satisfies the mean value property for all harmonic functions, then that domain is “almost” a ball. The goal of this note is to investigate some nonlocal counterparts of these results. Some of our arguments rely on fractional potential theory, others on purely nonlocal properties, with no classical counterpart, such as the fact that “all functions are locally fractional harmonic up to a small error”
Quasilinear logarithmic choquard equations with exponential growth in RN
Full text linkWe consider the N-Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential α is equal to the Euclidean dimension N, and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schrödinger energy is well defined is the Sobolev limiting space W1,N(RN), where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove the existence of finite energy solutions, we introduce a suitable log-weighted variant of the Pohozaev-Trudinger inequality which provides a suitable functional framework where we use variational methods
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