Potential Analysis最新文献

We consider a boundary value problem for the p-Laplacian, posed in the exterior of small cavities that all have the same p-capacity and are anchored to the unit sphere in (mathbb {R}^d), where (1<p<d.) We assume that the distance between anchoring points is at least (varepsilon ) and the characteristic diameter of cavities is (alpha varepsilon ), where (alpha =alpha (varepsilon )) tends to 0 with (varepsilon ). We also assume that anchoring points are asymptotically uniformly distributed as (varepsilon downarrow 0), and their number is asymptotic to a positive constant times (varepsilon ^{1-d}). The solution (u=u^varepsilon ) is required to be 1 on all cavities and decay to 0 at infinity. Our goal is to describe the behavior of solutions for small (varepsilon >0). We show that the problem possesses a critical window characterized by (tau :=lim _{varepsilon downarrow 0}alpha /alpha _c in (0,infty )), where (alpha _c=varepsilon ^{1/gamma }) and (gamma = frac{d-p}{p-1}.) We prove that outside the unit sphere, as (varepsilon downarrow 0), the solution converges to (A_*U) for some constant (A_*), where (U(x)=min {1,|x|^{-gamma }}) is the radial p-harmonic function outside the unit ball. Here the constant (A_*) equals 0 if (tau =0), while (A_*=1) if (tau =infty ). In the critical window where (tau ) is positive and finite, ( A_*in (0,1)) is explicitly computed in terms of the parameters of the problem. We also evaluate the limiting p-capacity in all three cases mentioned above. Our key new tool is the construction of an explicit ansatz function (u_{A_*}^varepsilon ) that approximates the solution (u^varepsilon ) in (L^{infty }(mathbb {R}^d)) and satisfies (Vert nabla u^varepsilon -nabla u_{A_*}^varepsilon Vert _{L^{p}(mathbb {R}^d)} rightarrow 0) as (varepsilon downarrow 0).

Abstract

We study random perturbations of a Riemannian manifold ((textsf{M},textsf{g})) by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields (h^bullet : omega mapsto h^omega ) will act on the manifold via the conformal transformation (textsf{g}mapsto textsf{g}^omega := e^{2h^omega },textsf{g}) . Our focus will be on the regular case with Hurst parameter (H>0) , the critical case  (H=0) being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

In this paper, we consider the following Schrödinger-Maxwell type equation with critical exponent (-Delta u=K(y)Big (frac{1}{|x|^{n-2}}*K(x)|u|^{frac{n+2}{n-2}}Big )u^{frac{4}{n-2}},quad {in},, mathbb {R}^n, qquad text {(0.1)}) where the function K satisfies the assumption (mathcal {F}), and (*) stands for the standard convolution. We first derived the non-degeneracy result for the critical Schrödinger-Maxwell equation. Then, as an application, we proved that problem Eq. (0.1) admits infinitely many non-radial positive solutions with arbitrary large energy. We believe that the various new ideas and technique computations that we used in this paper would be useful to deal with other related elliptic problems involving convolution nonlinear terms.

Abstract

We establish new characterizations of the Bloch space (mathcal {B}) which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function (f(z)=sum _{n=0}^infty widehat{f}(n) z^n) in the unit disc (mathbb {D}) , we define the fractional derivative ( D^{mu }(f)(z)=sum limits _{n=0}^{infty } frac{widehat{f}(n)}{mu _{2n+1}} z^n ) induced by a radial weight (mu ) , where (mu _{2n+1}=int _0^1 r^{2n+1}mu (r),dr) are the odd moments of (mu ) . Then, we consider the space ( mathcal {B}^mu ) of analytic functions f in (mathbb {D}) such that (Vert fVert _{mathcal {B}^mu }=sup _{zin mathbb {D}} widehat{mu }(z)|D^mu (f)(z)|<infty ) , where (widehat{mu }(z)=int _{|z|}^1 mu (s),ds) . We prove that (mathcal {B}^mu ) is continously embedded in (mathcal {B}) for any radial weight (mu ) , and (mathcal {B}=mathcal {B}^mu ) if and only if (mu in mathcal {D}=widehat{mathcal {D}}cap check{mathcal {D}}) . A radial weight (mu in widehat{mathcal {D}}) if (sup _{0le r<1}frac{widehat{mu }(r)}{widehat{mu }left( frac{1+r}{2}right) }<infty ) and a radial weight (mu in check{mathcal {D}}) if there exist (K=K(mu )>1) such that (inf _{0le r<1}frac{widehat{mu }(r)}{widehat{mu }left( 1-frac{1-r}{K}right) }>1.)

If the boundary of a domain in three dimensions is smooth enough, then the decay rate of the eigenvalues of the Neumann-Poincaré operator is known and it is optimal. In this paper, we deal with domains with less regular boundaries and derive quantitative estimates for the decay rates of the Neumann-Poincaré eigenvalues in terms of the Hölder exponent of the boundary. Estimates in particular show that the less the regularity of the boundary is, the slower is the decay of the eigenvalues. We also prove that the similar estimates in two dimensions. The estimates are not only for less regular boundaries for which the decay rate was unknown, but also for regular ones for which the result of this paper makes a significant improvement over known results.

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with (L^1) initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

On a smooth, closed Einstein manifold (M, g) of dimension (n ge 3) with positive scalar curvature and not conformally diffeomorphic to the standard sphere, we prove that the only conformal metrics to g with constant Q-curvature of order 4 are the metrics (lambda ) g with (lambda > 0) constant.

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems

where (a,epsilon >0), (lambda in mathbb {R}) is an unknown parameter that appears as a Lagrange multiplier, (V:mathbb {R}^N rightarrow [0,infty )) is a continuous function, and f is a continuous function with (L^2)-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential V attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations 4, 121–137 1996).

We consider the formal SDE

(textrm{d} X_t = b(t,X_t)textrm{d} t + textrm{d} Z_t, qquad X_0 = x in mathbb {R}^d, (text {E}))

where (bin L^r ([0,T],mathbb {B}_{p,q}^beta (mathbb {R}^d,mathbb {R}^d))) is a time-inhomogeneous Besov drift and (Z_t) is a symmetric d-dimensional (alpha )-stable process, (alpha in (1,2)), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, (L^r) and (mathbb {B}_{p,q}^beta ) respectively denote Lebesgue and Besov spaces. We show that, when (beta > frac{1-alpha + frac{alpha }{r} + frac{d}{p}}{2}), the martingale solution associated with the formal generator of (E) admits a density which enjoys two-sided heat kernel bounds as well as gradient estimates w.r.t. the backward variable. Our proof relies on a suitable mollification of the singular drift aimed at using a Duhamel-type expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.

Motivated by the notion of isotropic (alpha )-stable Lévy processes confined, by reflections, to a bounded open Lipschitz set (Dsubset mathbb {R}^d), we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time.