scholarly journals Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems with Rough Coefficients

2012 ◽  
Vol 64 (6) ◽  
pp. 1395-1414 ◽  
Author(s):  
Scott Rodney
Keyword(s):  
Functional Analysis ◽  
Quadratic Form ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Principal Part ◽  
Regularity Theory ◽  
Existence Theory ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Rough Coefficients

Abstract This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the formThe principal part ξ'P(x)ξ of the above equation is assumed to be comparable to a quadratic form Q(x,ξ)=ξ'Q(x)ξ that may vanish for non-zero ξ ∊ ℝn. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces QH1 (Θ)=W1,2(Θ,Q) and QH10(Θ)= W1,20 (Θ,Q)as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

scholarly journals Poincaré Inequalities and Neumann Problems for the p-Laplacian

2018 ◽  
Vol 61 (4) ◽  
pp. 738-753 ◽  
Author(s):  
David Cruz-Uribe ◽  
Scott Rodney ◽  
Emily Rosta
Keyword(s):  
Quadratic Form ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Poincaré Inequalities ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Poincare Inequalities ◽  
Associated Matrix ◽  
Weighted Poincaré Inequalities

AbstractWe prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.

15.—Differentiability on the Boundary of the Solutions of Linear Elliptic Boundary Value Problems

1975 ◽  
Vol 73 ◽  
pp. 235-249 ◽  
Author(s):  
Renate Schappel
Keyword(s):  
Functional Analysis ◽  
Boundary Value Problems ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Regularity Of Weak Solutions ◽  
Linear Elliptic Boundary

SynopsisThe present paper is concerned with the problem of regularity of weak solutions of boundary value problems. We shall present a new method to prove differentiability on the boundary. This method was developed in our thesis [12] within the theory of abstract Sobolev spaces, introduced by Stummel [14]. Here, we shall describe it by applying it to elliptic boundary value problems. It will be seen that the advantage of this method consists in the fact that it is based on functional analysis only and therefore may be used for other types of differential equations as well.

Existence Results for a Class of p(x)-Kirchhoff Problems

2021 ◽  
Vol 58 (1) ◽  
pp. 1-14
Author(s):  
Mostafa Allaoui
Keyword(s):  
Variational Methods ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Boundary Data ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Variable Exponent Sobolev Spaces ◽  
Robin Boundary

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhoff-type equations with Robin boundary data as follows:Where and satisfies Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

scholarly journals Quasilinear Dirichlet problems with competing operators and convection

2020 ◽  
Vol 18 (1) ◽  
pp. 1510-1517
Author(s):  
Dumitru Motreanu
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximation Procedure ◽  
Convection Term ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Variational Structure

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

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