Related papers: Remarks on the Rellich inequality
In this paper, we obtain some new results about weakly singular integral inequalities. These inequalities are used to discuss the global existence and uniqueness results for fractional differential equations of Riemann-Liouville type. Some…
We revisit the regularity theory for uniformly elliptic equations.
We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the…
We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of non-uniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses…
The aim of this note is to prove that almost-minimizers of the perimeter are Reifenberg flat, for a very weak notion of minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all…
Interpretability or explainability is an emerging research field in NLP. From a user-centric point of view, the goal is to build models that provide proper justification for their decisions, similar to those of humans, by requiring the…
The work presents two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula. These formulas involve a comparable number of parameters and provide non-Platonic…
Nonuniform ellipticity is a classical topic in the theory of partial differential equations. While several results in regularity theory have been adding up over decades, many basic issues, as for instance the validity of Schauder theory and…
Comparison results for solutions to the Dirichlet problems for a class of nonlinear, anisotropic parabolic equations are established. These results are obtained through a semi-discretization method in time after providing estimates for…
The paper concerns singular solutions of nonlinear elliptic equations.
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the…
In this paper, we establish some Stein-Weiss type inequalities with general kernels on the upper half space and study the existence of extremal functions for this inequality with the optimal constant. Furthermore, we also investigate the…
We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…
In this paper, we consider the Strichartz inequality for a fourth-order Schr\"odinger equation on $\mathbb{R}^{2+1}$. We show that extremizers exist using a linear profile decomposition which follows from the endpoint version decomposition…
We study the fair allocation of indivisible items under relevance constraints, where each agent has a set of relevant items and can only receive items that are relevant to them. While the relevance constraint has been studied in recent…
In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.
We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary…
In this article we consider mathematical fundamentals of one method for proving inequalities by computer, based on the Remez algorithm. Using the well-known results of undecidability of the existence of zeros of real elementary functions,…
We derive the exponential as well as power decreasing tail estimations for normed sums of centered independent identical distributed (or not) random variables on the Khintchine's form. We consider arbitrary, in particular, non-Rademacher's…
We obtain inequalities of H\"{o}lder and Minkowski type with weights generalizing both the case of weights with alternating signs and the classical case of non-negative weights.