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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 - div ( a ( x , u ) | | p - 2 u ) = λ b ( x , u ) | u | p - 2 u in Ω , u = 0 on Ω , 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 ( 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 𝔾 L n , the columns of the n × 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 B ( x 0 , d ) = { x n | x 0 - x | < d } , d < 1 2 , under the boundary-value conditions u ( x , k ) = k for x F , u ( x , k ) = 0 for x B ( x 0 , 1 ) and where 0 < ρ 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.