Related papers: Oscillating minimizers of a fourth order problem i…

We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique…

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R}…

This paper studies the oscillatory behavior of solutions to linear nonautonomous impulsive differential equations with piecewise constant arguments, including both advanced and delayed cases \[ x'(t) = a(t)x(t) + b(t)x([t-k]), \quad k \in…

The real anisotropic Littlewood's $4 / 3$ inequality is an extension of a famous result obtained in 1930 by J. E. Littlewood. It asserts that, for $a , b \in ( 0 , \infty )$, the following conditions are equivalent: $\bullet$ There is an…

The Calder\'on problem for the fractional Schr\"odinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a…

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication…

This paper considers a two-step fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving the time-variable fractional mobile-immobile advection-dispersion model subjects to suitable initial and boundary conditions.…

We present a new method to determine the best constant of the Sobolev-type embedding in one dimension with a norm including a bounded inhomogeneous potential term. This problem is closely connected to the Green function of the Schr\"odinger…

We consider minimization problems in the calculus of variations set in a sequence of domains the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain…

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\phi(u)$, with $\phi(u)=\Phi u^{3} + O(u^{5})$ odd and analytic, $\Phi\neq0$, and we construct small amplitude periodic solutions with frequency…

We prove that, for any measurable phase space subset $\Omega\subset\mathbb{R}^{2d}$ with $0<|\Omega|<\infty$ and any $1\le p < \infty$, the nonlinear concentration problem $$ \sup_{f \in…

The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the $p$-norm of the derivative among all functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero average.…

We prove existence and uniqueness of minimizers for a family of energy functionals that arises in Elasticity and involves polyconvex integrands over a certain subset of displacement maps. This work extends previous results by Awi and Gangbo…

We obtain new proofs with improved constants of the Khintchine-type inequality with matrix coefficients in two cases. The first case is the Pisier and Lust-Piquard noncommutative Khintchine inequality for $p=1$, where we obtain the sharp…

We take a qualitative comparative look at quantum and classical quartic anharmonic oscillators. It has been shown that the behavior of the quantum anharmonic oscillator mimics that of the classical anharmonic oscillators with the…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X \Delta^3(x){\rm d}x = BX^{7/4} + O_\epsilon(X^{\beta+\epsilon}) \qquad(B > 0) $$ and $$ \int_1^X…

While not generally a conservation law, any symmetry of the equations of motion implies a useful reduction of any second-order equationto a first-order equation between invariants, whose solutions (first integrals) can then be integrated by…

We consider an inverse extremal problem for variational functionals on arbitrary time scales. Using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variational functional that attains a…

Let $\alpha\in (0,2)$, let $${\cal E}(u,u)=\int_{\Bbb R^d}\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+\alpha}}\, dy\, dx$$ be the Dirichlet form for a stable-like operator, let $$\Gamma u(x)=\int_{\Bbb R^d}…

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…