We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, particular, fields $L^p$, $p>d$, as well some that are not even $L_{\loc}^{2+\varepsilon}$, $\varepsilon>0$. establish Hölder continuity of bounded soutions, sharp two-sided Gaussian bound on heat kernel, Harnack inequality.

We prove a boundary Harnack inequality for nonlocal elliptic operators L in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if Lu1 = Lu2 = 0 in Ω ∩ B1, u1 = u2 = 0 in B1 \ Ω, and u1, u2 ≥ 0 in R, then u1 and u2 are comparable in B1/2. The result applies to arbitrary open sets Ω. When Ω is Lipschitz, we show that the quotient u1/u2 is Hölder con...

The stability for the switched Cohen-Grossberg neural networks with mixed time delays and αinverse Hölder activation functions is investigated under the switching rule with the average dwell time property. By applying multiple Lyapunov-Krasovskii functional approach and linear matrix inequality LMI technique, a delay-dependent sufficient criterion is achieved to ensure such switched neural netw...

This paper introduces a new method to approximate Euclidean distances between points using block vectors in combination with the Hölder inequality. By defining lower bounds based on the proposed approximation, cluster algorithms can be considerably accelerated without loss of quality. In extensive experiments, we show a considerable reduction in terms of computational time in comparison to stan...

For p ∈ [2,∞) the metric Xp inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this result include the following sharp statements about embeddings of Lq into Lp when 2 < q < p <∞: the maximal θ ∈ (0, 1] for which Lq admits a bi-θ-Hölder embedding into Lp equals q/p, and for m,n ∈ N the smallest possible bi-Lipschitz distortion of any embeddi...

The stochastic optimal control uses the differential equation of Bellman and its solution—the Bellman function. We show how the homonym function in harmonic analysis is (and how it is not) the same stochastic optimal control Bellman function. Then we present several creatures from Bellman’s Zoo: a function that proves the inverse Hölder inequality, as well as several other harmonic analysis Bel...

We consider the system of stochastic differential equations dXt = A(Xt−) dZt, where Z t , . . . , Z d t are independent one-dimensional symmetric stable processes of order α, and the matrix-valued function A is bounded, continuous and everywhere non-degenerate. We show that bounded harmonic functions associated with X are Hölder continuous, but a Harnack inequality need not hold. The Lévy measu...

In this article, we consider the stochastic heat equation du = (∆u + f(t, x))dt + P∞ k=1 g(t, x)δβ t , t ∈ [0, T ], with random coefficients f and g, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of index H > 1/2. Using the Malliavin calculus techniques and a p-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to (βk)k, we prove that the equ...

Abstract The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection Jordan curves in Euclidean space. In the present paper we solve this setting proper metric spaces admitting local quadratic isoperimetric inequality for curves. We moreover obtain continuity up boundary and interior Hölder regularity solutions. Our results ge...

Abstract In this paper, we explore a class of Hermite–Hadamard integral inequalities for convex and m -convex functions. The Hölder inequality is used to create class, which has wide range applications in optimization theory. Some trapezoid-type midpoint error estimates are investigated. Inequalities several q -special functions highlighted. As particular cases, have included previous results.