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Observability inequalities and measurable sets

Jone Apraiz, Luis Escauriaza, Gengsheng Wang, C. Zhang (2014)

Journal of the European Mathematical Society

This paper presents two observability inequalities for the heat equation over $Ω \times (0, T)$ . In the first one, the observation is from a subset of positive measure in $Ω \times (0, T)$ , while in the second, the observation is from a subset of positive surface measure on $\partial Ω \times (0, T)$ . It also proves the Lebeau-Robbiano spectral inequality when $Ω$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

Observations on $W^{1, p}$ estimates for divergence elliptic equations with VMO coefficients

P. Auscher, M. Qafsaoui (2002)

Bollettino dell'Unione Matematica Italiana

In this paper, we make some observations on the work of Di Fazio concerning $W^{1, p}$ estimates, $1 < p < \infty$ , for solutions of elliptic equations $div A \nabla u = div f$ , on a domain $Ω$ with Dirichlet data $0$ whenever $A \in V M O (Ω)$ and $f \in L^{p} (Ω)$ . We weaken the assumptions allowing real and complex non-symmetric operators and $C^{1}$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose...

On 3D slightly compressible Euler equations.

Morando, Alessandro, Secchi, Paolo (2004)

Portugaliae Mathematica. Nova Série

On a Cahn–Hilliard model for phase separation with elastic misfit

Harald Garcke (2005)

Annales de l'I.H.P. Analyse non linéaire

On a class of initial-boundary value problems in a domain with boundary containing isolated points

H. Marcinkowska (1983)

Annales Polonici Mathematici

On a class of multitime evolution equations with nonlocal initial conditions.

Zouyed, F., Rebbani, F., Boussetila, N. (2007)

Abstract and Applied Analysis

On a class of nonlinear reaction-diffusion systems with nonlocal boundary conditions.

Bouziani, Abdelfatah (2004)

Abstract and Applied Analysis

On a class of singular hyperbolic equation with a weighted integral condition.

Mesloub, Said, Bouziani, Abdelfatah (1999)

International Journal of Mathematics and Mathematical Sciences

On a conjugate semi-variational method for parabolic equations

Ivan Hlaváček (1973)

Aplikace matematiky

On a conserved Penrose-Fife type system

Gianni Gilardi, Andrea Marson (2005)

Applications of Mathematics

We deal with a class of Penrose-Fife type phase field models for phase transitions, where the phase dynamics is ruled by a Cahn-Hilliard type equation. Suitable assumptions on the behaviour of the heat flux as the absolute temperature tends to zero and to $+ \infty$ are considered. An existence result is obtained by a double approximation procedure and compactness methods. Moreover, uniqueness and regularity results are proved as well.

On a Darboux problem for a third-order hyperbolic equation with multiple characteristics.

Jokhadze, O. (1995)

Georgian Mathematical Journal

On a Differential Inequality and Its Applications to Geometry.

Catherine Bandle (1976)

Mathematische Zeitschrift

On a general class of elliptic equations in devergence.

Giovanni Gregori (1993)

Mathematische Zeitschrift

On a generalized nonlinear equation of Schrödinger type.

Feng, Xueshang (1995)

Bulletin of the Belgian Mathematical Society - Simon Stevin

On a Global Existence Theorem of Small Amplitude Solutions for Nonlinear Wave Equations in an Exterior Domain.

Yoshihiro Shibata, Yoshio Tsutsumi (1986)

Mathematische Zeitschrift

On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long (2024)

Mathematica Bohemica

We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u (η_{1}, t), \dots, u (η_{q}, t)$ with $0 \leq η_{1} < η_{2} < \dots < η_{q} < 1 .$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $(P_{q})$ of (P) in which the nonlinear term contains the sum $S_{q} [u^{2}] (t) = q^{- 1} \sum_{i = 1}^{q} u^{2} (\frac{(i - 1)}{q}, t)$ . Under suitable conditions, we prove that the solution of $(P_{q})$ converges to the solution of the corresponding...

On a nonlocal eigenvalue problem

Juncheng Wei, Liqun Zhang (2001)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball

Patrick Oswald (1985)

Commentationes Mathematicae Universitatis Carolinae

On a singular sublinear polyharmonic problem.

Ben Othman, Sonia (2006)

Abstract and Applied Analysis

On a Singular Value Problem for the Fractional Laplacian on the Exterior of the Unit Ball

Bezzarga, Mounir, Kefi, Khaled (2005)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: Primary 26A33; Secondary 47G20, 31B05We study a singular value problem and the boundary Harnack principle for the fractional Laplacian on the exterior of the unit ball.

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