A second look on definition and equivalent norms of Sobolev spaces

Joachim Naumann; Christian G. Simader

Displaying similar documents to “A second look on definition and equivalent norms of Sobolev spaces”

The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems

Pavel Drábek (1995)

Mathematica Bohemica

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We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem $\begin{matrix} - div (a (x, u) | |^{p - 2} \nabla u) = & {λ b (x, u) | u |}^{p - 2} u in Ω, u = & 0 on \partial Ω, \end{matrix}$ where $Ω$ is a bounded domain, $p > 1$ is a real number and $a (x, u)$ , $b (x, u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a (x, u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^{\infty} (Ω)$ . The main tool is the investigation of the associated homogeneous eigenvalue problem...

Linear Stieltjes integral equations in Banach spaces

Štefan Schwabik (1999)

Mathematica Bohemica

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Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in []. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. []). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the...

Linear integral equations in the space of regulated functions

Milan Tvrdý (1998)

Mathematica Bohemica

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n this paper we investigate systems of linear integral equations in the space $𝔾_{L}^{n}$ of $n$ -vector valued functions which are regulated on the closed interval $[0, 1]$ (i.e. such that can have only discontinuities of the first kind in $[0, 1]$ ) and left-continuous in the corresponding open interval $(0, 1) .$ In particular, we are interested in systems of the form x(t) - A(t)x(0) - 01B(t,s)[d x(s)] = f(t), where $f \in 𝔾_{L}^{n}$ , the columns of the $n \times n$ -matrix valued function $A$ belong to $𝔾_{L}^{n}$ , the entries of $B (t, .)$ have a bounded variation...

On weighted estimates of solutions of nonlinear elliptic problems

Igor V. Skrypnik, Dmitry V. Larin (1999)

Mathematica Bohemica

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The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, $2 p < n$ for a solution of a degenerate nonlinear elliptic equation in a domain $B (x_{0}, 1) ∖ F$ , $F \subset B (x_{0}, d) = {x \in ℝ^{n} | x_{0} - x | < d}$ , $d < \frac{1}{2}$ , under the boundary-value conditions $u (x, k) = k$ for $x \in \partial F$ , $u (x, k) = 0$ for $x \in \partial B (x_{0}, 1)$ and where $0 < ρ \leq d i s t (x, F)$ , $w (x)$ is a weighted function from some Muckenhoupt class, and $c a p_{p, w} (F)$ , $w (B (x, ρ))$ are weighted capacity and measure of the corresponding sets.

The obstacle problem for functions of least gradient

William P. Ziemer, Kevin Zumbrun (1999)

Mathematica Bohemica

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For a given domain $Ω \subset ℝ^{n}$ , we consider the variational problem of minimizing the $L^{1}$ -norm of the gradient on $Ω$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower obstacle condition $u \geq ψ$ inside $Ω$ . Under the assumption of strictly positive mean curvature of the boundary $\partial Ω$ , we show existence of a continuous solution, with Holder exponent half of that of data and obstacle. This generalizes previous results obtained for the unconstrained and double-obstacle...

Killing's equations in dimension two and systems of finite type

Gerard Thompson (1999)

Mathematica Bohemica

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A PDE system is said to be of finite type if all possible derivatives at some order can be solved for in terms lower order derivatives. An algorithm for determining whether a system of finite type has solutions is outlined. The results are then applied to the problem of characterizing symmetric linear connections in two dimensions that possess homogeneous linear and quadratic integrals of motions, that is, solving Killing's equations of degree one and two.