scholarly journals A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Paolo Secchi
Keyword(s):  
Sobolev Spaces ◽  
Hyperbolic Systems ◽  
Type Inequality ◽  
Initial Boundary ◽  
Higher Order ◽  
Hardy Type Inequality ◽  
Characteristic Boundary ◽  
Hardy Type ◽  
Anisotropic Sobolev Spaces ◽  
Initial Boundary Value

We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.

THE TRACE THEOREMS IN ANISOTROPIC SOBOLEV SPACES AND THEIR APPLICATIONS TO THE CHARACTERISTIC INITIAL BOUNDARY VALUE PROBLEM FOR SYMMETRIC HYPERBOLIC SYSTEMS

1995 ◽  
Vol 05 (08) ◽  
pp. 1079-1092 ◽  
Author(s):  
YASUSHI SHIZUTA ◽  
KÔZÔ YABUTA
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Trace Theorems ◽  
Characteristic Boundary ◽  
Anisotropic Sobolev Spaces ◽  
Initial Boundary Value

Anisotropic Sobolev spaces are introduced in order to study the initial boundary value problem for first-order symmetric hyperbolic systems with characteristic boundary of constant multiplicity. A trace theorem is given and used for showing the necessity of the compatibility condition for the existence of solution that lies in the anisotroropic Sobolev space.

A Sharp Adams-Type Inequality for Weighted Sobolev Spaces

2020 ◽  
Vol 71 (2) ◽  
pp. 517-538
Author(s):  
João Marcos do Ó ◽  
Abiel Costa Macedo ◽  
José Francisco de Oliveira
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Sobolev Type ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Classical Work

Abstract In a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

scholarly journals Characterizations of weighted dynamic Hardy-type inequalities with higher-order derivatives

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. H. Saker ◽  
R. R. Mahmoud ◽  
K. R. Abdo
Keyword(s):  
Time Scales ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Hardy Type Inequality ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Weighted Functions ◽  
Necessary And Sufficient

AbstractIn this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when $\mathbb{T=R}$ T = R and $\mathbb{T=N}$ T = N , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new.

scholarly journals Three weights higher order Hardy type inequalities

2006 ◽  
Vol 4 (2) ◽  
pp. 163-191
Author(s):  
Aigerim A. Kalybay ◽  
Lars-Erik Persson
Keyword(s):  
Differential Operator ◽  
Weight Function ◽  
Type Inequality ◽  
Higher Order ◽  
Hardy Type Inequality ◽  
Volterra Type ◽  
Crucial Step ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Volterra Type Operator

We investigate the following three weights higher order Hardy type inequality (0.1)‖g‖q,u≤  C‖Dρkg‖p,vwhereDρidenotes the following weighted differential operator:{dig(t)dti,i=0,1,...,m−1,di−mdti−m(p(t)dmg(t)dtm),i=m,m+1,...,k,for a weight functionρ(⋅). A complete description of the weightsu,vandρso that (0.1) holds was given in [4] for the case1<p≤q<∞. Here the corresponding characterization is proved for the case1<q<p<∞. The crucial step in the proof of the main result is to use a new Hardy type inequality (for a Volterra type operator), which we first state and prove.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

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