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This survey on stationary and evolutionary problems with gradient constraints is based on developments of monotonicity and compactness methods applied to large classes of scalar and vectorial solutions to variational and quasi-variational…
This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a…
We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional…
This paper was devoted to study the quantitative homogenization problems for nonlinear elliptic operators in perforated domains. We obtained a sharp error estimate $O(\varepsilon)$ when the problem was anchored in the reference domain…
The effectiveness of non-parametric, kernel-based methods for function estimation comes at the price of high computational complexity, which hinders their applicability in adaptive, model-based control. Motivated by approximation techniques…
We investigate the fractional Orlicz boundary Hardy-type inequality for bounded Lipschitz domains. Further, we establish fractional Orlicz boundary Hardy-type inequalities with logarithmic corrections for specific critical cases across…
We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive…
Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary…
We construct a non - improved exponential bounds for distribution of normed sums of i.,i.d. random variables with random numbers of summand.
Mixed trigonometric-polynomials frequently occur in applications in physics, numerical analysis and engineering, the algorithm has been already proposed to determine its sign on (0,{pi}/2]. This paper proposes a procedure to extend the…
In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential…
In the present paper we find optimal conditions separating the regular case from the one with Lavrentiev gap for the borderline case of double phase potencial and related general classes of integrands. We present new results on density of…
The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems:…
We provide sufficient conditions for the Lojasiewicz-Simon gradient inequality to hold on a submanifold of a Banach space and discuss the optimality of our assumptions. Our result provides a tool to study asymptotic properties of…
We provide error bounds on the traces and norms of the derivative of the $L^2$ projection of an $H^{k}$ function onto the space of polynomials of degree $\leq p$. The bounds are explicit in the order of differentiation and the polynomial…
A plethora of spaces in Functional Analysis (Braun-Meise-Taylor and Carleman ultradifferentiable and ultraholomorphic classes; Orlicz, Besov, Lipschitz, Lebesque spaces, to cite the main ones) are defined by means of a weighted structure,…
We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also…
We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts…
A new characterization of the exponential type Orlicz spaces generated by the functions $\exp(|x|^p)-1$ ($p\ge 1$) is given. We define norms for centered random variables belonging to these spaces. We show equivalence of these norms with…
We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at…