Potential Analysis最新文献

We prove the monotonicity of positive solutions to the equation (-Delta _p u = f(u)) in (mathbb {R}^N_+) with zero Dirichlet boundary condition, where (1<p<2) and (f:[0,+infty )rightarrow mathbb {R}) is a continuous function which is positive and locally Lipschitz continuous in ((0,+infty )) and (liminf _{trightarrow 0^+}frac{f(t)}{t^{p-1}}>0). Furthermore, we allow f to be sign-changing in the case (frac{2N+2}{N+2}<p<2). The celebrated moving plane method will be used in the proofs of our results.

It is known that on RCD spaces one can define a distributional Ricci tensor (textbf{Ric}). Here we give a fine description of this object by showing that it admits the polar decomposition

for a suitable non-negative measure (|textbf{Ric}|) and unitary tensor field (omega ). The regularity of both the mass measure and of the polar vector are also described. The representation provided here allows to answer some open problems about the structure of the Ricci tensor in such singular setting. Our discussion also covers the case of Hessians of convex functions and, under suitable assumptions on the base space, of the Sectional curvature operator.

Long-time behavior of diffusion processes with one-sided random potentials starting from the origin is studied. As random potentials, some strictly stable processes are given just on the negative side in the real line. This model is an extension of the diffusion process with a one-sided Brownian potential studied by Kawazu, Suzuki and Tanaka (Tokyo J. Math. 24, 211–229 2001) and Kawazu and Suzuki (J. Appl. Probab. 43, 997–1012 2006). In this paper, we analyze our model by different methods from theirs. We use the theory concerning the convergence of a sequence of bi-generalized diffusion processes studied by Ogura (J. Math. Soc. Japan 41, 213–242 1989) and Tanaka (Comm. Pure Appl. Math. 47, 755–766 1994). For diffusion processes with one-sided random potentials, the limit theorems introduced by them cannot be used. We improve their limit theorems and apply the improved limit theorem to examining the long-time behavior of our model. As a result, we show that limit distributions exist under the Brownian scaling with some probability, and under a sub-diffusive scaling with the remaining probability.

Let (L^p(textbf{T})) be the Lesbegue space of complex-valued functions defined in the unit circle (textbf{T}={z: |z|=1}subseteq mathbb {C}). In this paper, we address the problem of finding the best constant in the inequality of the form:

Here (pin [1,2]), (b>0), and by (P_{-} f) and ( P_+ f) are denoted the co-analytic and analytic projections of a function (fin L^p(textbf{T})). The sharpness of the constant (A_{p,b}) follows by taking a family quasiconformal harmonic mapping (f_c) and letting (crightarrow 1/p). The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.

In this paper, we establish a large deviation principle for the solutions to the stochastic heat equations with logarithmic nonlinearity driven by Brownian motion, which is neither locally Lipschitz nor locally monotone. Nonlinear versions of Gronwall’s inequalities and Log-Sobolev inequalities play an important role.

We investigate the well-posedness of following McKean-Vlasov equation in (mathbb {R}^d):

where (mu _{X_t}) is the law of (X_t). The existence of solutions is demonstrated when (sigma ) satisfies certain non-degeneracy and continuity assumptions, and when b meets some integrability conditions, and continuity requirements in the (generalized) total variation distance. Furthermore, uniqueness is established under additional continuity assumptions of a Lipschitz type.

Given a compact doubling metric measure space X that supports a 2-Poincaré inequality, we construct a Dirichlet form on (N^{1,2}(X)) that is comparable to the upper gradient energy form on (N^{1,2}(X)). Our approach is based on the approximation of X by a family of graphs that is doubling and supports a 2-Poincaré inequality (see [20]). We construct a bilinear form on (N^{1,2}(X)) using the Dirichlet form on the graph. We show that the (Gamma )-limit (mathcal {E}) of this family of bilinear forms (by taking a subsequence) exists and that (mathcal {E}) is a Dirichlet form on X. Properties of (mathcal {E}) are established. Moreover, we prove that (mathcal {E}) has the property of matching boundary values on a domain (Omega subseteq X). This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form (mathcal {E})) on a domain in X with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.

We establish the Lipschitz regularity of the a priori bounded local minimizers of integral functionals with non autonomous energy densities satisfying non standard growth conditions under a bound on the gap between the growth and the ellipticity exponent that is reminiscent of the sharp bound already found in [16].

The asymptotic analysis of a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients covering a large variety of physical systems, a wide class of quasilinear SPDEs and a good number of fluid dynamic models is carried out in this work. The aim of this work is to develop the large deviation theory for small Gaussian as well as Poisson noise perturbations of the above class of SPDEs. We establish a Wentzell-Freidlin type large deviation principle for the strong solution to such SPDEs perturbed by a multiplicative Lévy noise in a suitable Polish space using a variational representation (based on a weak convergence approach) for nonnegative functionals of general Poisson random measures and Brownian motions. The well-posedness of an associated deterministic control problem is established by exploiting pseudo-monotonicity arguments and the stochastic counterpart is obtained by an application of Girsanov’s theorem.

If ((mathcal{E}, mathcal{D})) is a symmetric, regular, strongly local Dirichlet form on (L^2 (X,m)), admitting a carré du champ operator (Gamma ), and (p>1) is a real number, then one can define a nonlinear form (mathcal{E}^p) by the formula

where u, v belong to an appropriate subspace of the domain (mathcal{D}). We show that (mathcal{E}^p) is a nonlinear Dirichlet form in the sense introduced by P. van Beusekom. We then construct the associated Choquet capacity. As a particular case we obtain the nonlinear form associated with the p-Laplace operator on (W_0^{1,p}). Using the above procedure, for each n-dimensional quasiregular mapping f we construct a nonlinear Dirichlet form (mathcal{E}^n) ((p=n)) such that the components of f become harmonic functions with respect to (mathcal{E}^n). Finally, we obtain Caccioppoli type inequalities in the intrinsic metric induced by (mathcal{E}), for harmonic functions with respect to the form (mathcal{E}^p).