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Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta, Anibal Rodriguez-Bernal, Philippe Souplet (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions u : either the space derivative u x blows up in finite time (with u itself remaining bounded), or u is global and converges in C 1 norm to the unique steady state. The main difficulty is to prove C 1 boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...

Boundedness of the solution of the third problem for the Laplace equation

Dagmar Medková (2005)

Czechoslovak Mathematical Journal

A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.

Bounding the degree of solutions to Pfaff equations

Marco Brunella, Luis Gustavo Mendes (2000)

Publicacions Matemàtiques

We study hypersurfaces of complex projective manifolds which are invariant by a foliation, or more generally which are solutions to a Pfaff equation. We bound their degree using classical results on logarithmic forms.

Bounds and estimates on the effective properties for nonlinear composites

Peter Wall (2000)

Applications of Mathematics

In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle...

Bounds and numerical results for homogenized degenerated p -Poisson equations

Johan Byström, Jonas Engström, Peter Wall (2004)

Applications of Mathematics

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p -Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.

Bounds for KdV and the 1-d cubic NLS equation in rough function spaces

Herbert Koch (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time H s bounds in terms of the H s size of the initial data for s ≥ - 1 4 (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in H - 1 (joint work with T. Buckmaster [2]).

Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

Pedro Freitas, Batłomiej Siudeja (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improve Pólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extend Hersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals.

Bounds of Riesz Transforms on L p Spaces for Second Order Elliptic Operators

Zhongwei Shen (2005)

Annales de l’institut Fourier

Let ℒ = -div ( A ( x ) ∇ ) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ℝ n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2 , a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇ ( ℒ ) - 1 / 2 on the L p space. As an application, for 1 < p < 3 + ϵ , we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V M O coefficients. The range of p is sharp. The closely related boundedness of ...

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