The Dirichlet problem for the Laplace equation for a planar domain with piecewise-smooth boundary is studied using the indirect integral equation method. The domain is bounded or unbounded. It is not supposed that the boundary is connected. The boundary conditions are continuous or p-integrable functions. It is proved that a solution of the corresponding integral equation can be obtained using the successive approximation method.

Inequalities concerning the integral of |∇u|2 on the subsets where |u(x)| is greater than k can be used in order to prove regularity properties of the function u. This method was introduced by Ennio De Giorgi e Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems.

The paper investigates the third boundary value problem $\frac{\partial u}{\partial n}+\lambda u=\mu$ for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where $\nu$ is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure $T\nu$. Denote by $T\nu \to T\nu$ the corresponding operator on the space of signed measures on the boundary of the investigated domain $G$. If there is $\alpha \ne 0$ such that the essential spectral radius of $(\alpha I-T)$ is...

We show that the spectrum of $\Delta {²}_{p}u+{2\beta ·\nabla \left(\right|\Delta u|}^{p-2}{\Delta u)+|\beta \left|²\right|\Delta u|}^{p-2}\Delta u=\alpha m{\left|u\right|}^{p-2}u$, where $\beta \in {ℝ}^N$, under Navier boundary conditions, contains at least one sequence of eigensurfaces.

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional...

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional...

Suppose $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^n$, which is nonincreasing in $\left|x\right|$. Set $\left(Tf\right)\left(x\right)={\int }_{R^n}\Phi (x-y)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^n$. Given $1\<p\<\infty$ and $v\ge 0$, we show there exists $C\>0$ so that ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ for all $f\ge 0$, if and only if $C^{\text{'}}\>0$ exists with ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\text{'}}}dx\le C^{\text{'}}{\int }_{Q}v\left(x\right)dx\<\infty$ for all dyadic cubes Q, where $p^{\text{'}}=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

The unique solvability of the problem Δu = 0 in G⁺ ∪ G¯, u₊ - au_ = f on ∂G⁺, n⁺·∇u₊ - bn⁺·∇u_ = g on ∂G⁺ is proved. Here a, b are positive constants and g is a real measure. The solution is constructed using the boundary integral equation method.

In this note we study the waiting time phenomenon for local solutions of the nonlinear diffusion equation through its connection with the nondiffusion of the support property for local solutions of the family of discretized elliptic problems. We show that, under a suitable growth condition on the initial datum near the boundary of its support, a finite part of the family of solutions of the discretized problem maintain unchanged its support.

We consider degenerated elliptic equations of the form$$\sum _{i,j}{D}_{x_i}(a_{ij}\left(x\right)D_{x_j}),\text{where}\lambda w(x\left)\right|\xi {|}^2\le \sum _{i,j}{a}_{ij}\left(x\right){\xi }_i{\xi }_j\le \Lambda w\left(x\right)|\xi {|}^2.$$Under suitable assumptions on $w$, we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on $w$. The main tool we use is an estimate for the Green function in terms of $w$.