Related papers: Boundary value problem and the Ehrhard inequality

In this note, we study a quantitative extension of the John-Nirenberg inequality for the Hardy-Littlewood maximal function of a $\operatorname{BMO}$ function. More precisely, for every nonconstant locally integrable function $f$ such that…

In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous…

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ with $\theta >…

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In…

In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel \begin{equation*} \int_{\mathbb{R}_+^n}\int_{\partial\mathbb{R}^n_+} \frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x) dydx\geq…

The boundary value problem is examined for the system of elliptic equations of from $-\Delta u + A(x)u = 0 \quad\text{in} \Omega,$ where $A(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$ and $\frac{\partial u}{\partial…

We investigate elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on…

We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\Delta^y(E) = \{|x-y|:x\in E\},$$ we prove that for any $E, F\subset{\Bbb R}^d$, there exists a…

We employ a variational approach to study the Neumann boundary value problem for the $p$-Laplacian on bounded smooth-enough domains in the metric setting, and show that solutions exist and are bounded. The boundary data considered are Borel…

The logarithmic Sobolev inequality for the Hamming cube {0,1}^n states that for any real-valued function f on the cube holds E(f,f) \ge 2 Ent(f^2), where E(f,f) is the appropriate Dirichlet form (also known as "sum of influences"). We show…

\begin{abstract} In this paper we state the following weighted Hardy type inequality for any functions $\varphi$ in a weighted Sobolev space and for weight functions $\mu$ of a quite general type \begin{equation*} c_{N,\mu}…

We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - \theta)$ over the $n$-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every…

Let $(\Sigma,g)$ be a compact Riemann surface with smooth boundary $\partial\Sigma$, $\Delta_g$ be the Laplace-Beltrami operator, and $h$ be a positive smooth function. Using a min-max scheme introduced by Djadli-Malchiodi (2006) and Djadli…

Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $\mu$ be the M\"obius function and $\mu_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=\mu^2g$ and $f=\mu_2^2g$ have unbounded partial sums. Our proofs…

In this paper, the connection between the functional inequalities $$ f\Big(\frac{x+y}{2}\Big)\leq\frac{f(x)+f(y)}{2}+\alpha_J(x-y) \qquad (x,y\in D)$$ and $$ \int_0^1f\big(tx+(1-t)y\big)\rho(t)dt \leq\lambda f(x)+(1-\lambda)f(y)…

We are concerned with positive solutions of equation (E) $(-\Delta)^s u=f(u)$ in a domain $\Omega \subset \mathbb{R}^N$ ($N>2s$), where $s \in (\frac{1}{2},1)$ and $f\in C^{\alpha}_{loc}(\mathbb{R})$ for some $\alpha \in(0,1)$. We establish…

If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are non-negative measurable functions, then the Pr\'ekopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater or equal than the…

Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ \mu $ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present…

In the paper we prove a generalization of the Hopf lemma for weak subsolutions of the equation: $-Au+cu=0$ in $D$, for a wide class of L\'evy type integro-differential operators $A$, bounded and measurable function $c:D\to[0,+\infty)$ and…

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and…