Potential Analysis最新文献

We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

We consider second order degenerate parabolic equations with real, measurable, and time-dependent coefficients. We allow for degenerate ellipticity dictated by a spatial (A_2)-weight. We prove the existence of a fundamental solution and derive Gaussian bounds. Our construction is based on the original work of Kato (Nagoya Math. J. 19, 93–125 1961).

In this paper we prove regularity result for solutions of the boundary value problem

with the vector field E(x) and the function f(x) belonging to some Marcinkiewicz spaces.

We examine the relations between different capacities in the setting of a metric measure space. First, we prove a comparability result for the Riesz ((beta ,p))-capacity and the relative Hajłasz ((beta ,p))-capacity, for (1<p<infty ) and (0<beta le 1), under a suitable kernel estimate related to the Riesz potential. Then we show that in geodesic spaces the corresponding capacity density conditions are equivalent even without assuming the kernel estimate. In the last part of the paper, we compare the relative Hajłasz (1, p)-capacity to the relative variational p-capacity.

We prove the intrinsic Harnack’s inequality for a general form of a parabolic equation that generalizes both the standard parabolic p-Laplace equation and the normalized version arising from stochastic game theory. We prove each result for the optimal range of exponents and ensure that we get stable constants.

We found the exact asymptotics of the singular numbers for the Cauchy transform and its product with Bergman’s projection over the space (L^{2}(Omega ),) where (Omega ) is a doubly-connected domain in the complex plane.

Embeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.

We prove non-uniqueness in law of the three-dimensional magnetohydrodynamics system that is forced by random noise of an additive and a linear multiplicative type and has viscous and magnetic diffusion, both of which are weaker than a full Laplacian. We apply convex integration to both equations of velocity and magnetic fields in order to obtain the non-uniqueness in law in the class of probabilistically strong solutions.

On weighted Riemannian manifolds we prove the existence of globally Lipschitz transport maps between the weight (probability) measure and log-Lipschitz perturbations of it, via Kim and Milman’s diffusion transport map, assuming that the curvature-dimension condition (varvec{textrm{CD}(rho _{1}, infty )}) holds, as well as a second order version of it, namely (varvec{Gamma _{3} ge rho _{2} Gamma _{2}}). We get new results as corollaries to this result, as the preservation of Poincaré’s inequality for the exponential measure on (varvec{(0,+infty )}) when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on (varvec{(0,+infty )}) (then getting as a particular case the exponential one), via Laguerre’s generator.

In the setting of a metric space equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain (Omega ) of functions bounded between two obstacle functions inside (Omega ), and whose trace lies between two prescribed functions on the boundary. If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when (Omega ) is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Since candidate functions need not agree outside of the domain, standard compactness arguments fail to provide existence of weak solutions as they are defined for the problem with single boundary condition. To overcome this, we introduce a class of ( varepsilon )-weak solutions as an intermediate step. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.