This paper deals with tangential boundary behaviors of harmonic functions with gradient in Lebesgue classes. Our aim is to extend a recent result of Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), concerning tangential boundary limits of harmonic functions with gradient in $L^n({\mathbf{R}}_{+}^n)$, ${\mathbf{R}}_{+}^n$ denoting the upper half space of the $n$-dimensional euclidean space ${\mathbf{R}}^n$. Our method used here is different from that of Nagel, Rudin and Shapiro (Ann. of Math., 116 (1982), 331–360); in fact, we use the integral representation...

In this paper we deal with the stationary Navier-Stokes problem in a domain $\Omega$ with compact Lipschitz boundary $\partial \Omega$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial \Omega$, with possible countable exceptional set, provided $a$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $\Omega$ bounded.

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and a convergence...

In this paper, we give an overview of some topics involving behavior of homeomorphisms and ways in which real analysis can arise in geometric settings.

We consider holomorphic functions and complex harmonic functions on some balls, including the complex Euclidean ball, the Lie ball and the dual Lie ball. After reviewing some results on Bergman kernels and harmonic Bergman kernels for these balls, we consider harmonic continuation of complex harmonic functions on these balls by using harmonic Bergman kernels. We also study Szegő kernels and harmonic Szegő kernels for these balls.

Extending recent work for the linear Poisson problem for the Laplacian in the framework of Sobolev-Besov spaces on Lipschitz domains by Jerison and Kenig [16], Fabes, Mendez and Mitrea [9], and Mitrea and Taylor [30], here we take up the task of developing a similar sharp theory for semilinear problems of the type Δu - N(x,u) = F(x), equipped with Dirichlet and Neumann boundary conditions.

We characterize all subsets $M$ of ${ℝ}^n\times {ℝ}^{+}$ such that $$\underset{X\in {ℝ}^n\times {ℝ}^{+}}{sup}u\left(X\right)=\underset{X\in M}{sup}u\left(X\right)$$ for every bounded parabolic function $u$ on ${ℝ}^n\times {ℝ}^{+}$. The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points of $M$ is also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated.

Let $\alpha >0$, $\lambda ={\left(2\alpha \right)}^{-1/2}$, $S^{n-1}$ be the $(n-1)$-dimensional unit sphere, $\sigma$ be the surface measure on $S^{n-1}$ and $h\left(x\right)={\int }_{S^{n-1}}{e}^{\lambda \langle x,y\rangle }d\sigma \left(y\right)$. We characterize all subsets $M$ of ${ℝ}^n$ such that $$\underset{x\in {ℝ}^n}{inf}\frac{u\left(x\right)}{h\left(x\right)}=\underset{x\in M}{inf}\frac{u\left(x\right)}{h\left(x\right)}$$ for every positive solution $u$ of the Helmholtz equation on ${ℝ}^n$. A closely related problem of representing functions of $L_1\left(S^{n-1}\right)$ as sums of blocks of the form $e^{\lambda \langle x_k,.\rangle }/h\left(x_k\right)$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.

For open sets with a piecewise smooth boundary it is shown that a solution of the Dirichlet problem for the Laplace equation can be expressed in the form of the sum of the single layer potential and the double layer potential with the same density, where this density is given by a concrete series.

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.