Dans cet article, on étudie la régularité d’une solution réelle, appartenant à $H^s$ pour $s$ assez grand, d’une équation aux dérivées partielles strictement hyperbolique et fortement non linéaire d’ordre deux. On suppose que les données de Cauchy sur une hypersurface spatiale lisse sont régulières en dehors d’un point, et ont une singularité conormale en ce point; on démontre alors que la réunion $\Gamma$ des bicaractéristiques nulles issues de ce point est, en dehors de ce point, une hypersurface lisse et...

Sufficient conditions for the stresses in the threedimensional linearized coupled thermoelastic system including viscoelasticity to be continuous and bounded are derived and optimization of heating processes described by quasicoupled or partially linearized coupled thermoelastic systems with constraints on stresses is treated. Due to the consideration of heating regimes being “as nonregular as possible” and because of the well-known lack of results concerning the classical regularity of solutions...

Si dimostra resistenza e l'unicità della soluzione del problema $Au=f$, $u\in H_0^1(\mathrm{\Omega })$ nel caso in cui $\mathrm{\Omega }$ è un aperto di ${ℝ}^n$ non limitato, $A$ è un operatore variazionale ellittico del secondo ordine a coefficienti misurabili e limitati e $f$ appartiene a $H^{-1}(\mathrm{\Omega })$.

We consider the general degenerate parabolic equation :$$u_t-\Delta b\left(u\right)+div\tilde{F}\left(u\right)=f\text{in}Q=]0,T[\times {ℝ}^N,T\>0.$$We suppose that the flux $\tilde{F}$ is continuous, $b$ is nondecreasing continuous and both functions are not necessarily Lipschitz. We prove the existence of the renormalized solution of the associated Cauchy problem for $L^1$ initial data and source term. We establish the uniqueness of this type of solution under a structure condition $\tilde{F}\left(r\right)=F\left(b\left(r\right)\right)$ and an assumption on the modulus of continuity of $b$. The novelty of this work is that $\Omega ={ℝ}^N$, $u_0$, $f\in L^1$, $b$, $\tilde{F}$ are not Lipschitz...

In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable $\theta$-scheme with $1/2\le \theta \le 1$. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization...

In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization...