Generalized Wentzell boundary conditions for second order operators with interior degeneracy
Genni Fragnelli Gisèle Ruiz Goldstein Jerome Goldstein Rosa Maria Mininni Silvia Romanelli
Discrete & Continuous Dynamical Systems - S 2016, 9(3): 697-715 doi: 10.3934/dcdss.2016023
We consider operators in divergence form, $A_1u=(au')'$, and in nondivergence form, $A_2u=au''$, provided that the coefficient $a$ vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators $A_1$ and $A_2$, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of $L^2$ type.
keywords: Second order operators in divergence and nondivergence form interior degeneracy generalized Wentzell boundary conditions.
A Cahn-Hilliard-Gurtin model with dynamic boundary conditions
Gisèle Ruiz Goldstein Alain Miranville
Discrete & Continuous Dynamical Systems - S 2013, 6(2): 387-400 doi: 10.3934/dcdss.2013.6.387
Our aim in this paper is to define proper dynamic boundary conditions for a generalization of the Cahn-Hilliard system proposed by M. Gurtin. Such boundary conditions take into account the interactions with the walls in confined systems. We then study the existence and uniqueness of weak solutions.
keywords: Cahn-Hilliard-Gurtin equations dynamic boundary conditions well-posedness.
Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces
Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Silvia Romanelli
Discrete & Continuous Dynamical Systems - S 2011, 4(3): 581-593 doi: 10.3934/dcdss.2011.4.581
Let us consider the operator $A_n u$:=$(-1)^{n+1} \alpha (x) u^{(2n)}$ on $H^n_0(0,1)$ with domain $D(A_n)$:=$\{u\in H^n_0(0,1)\cap H^{2n}$loc$(0,1)\ :\ A_n u\in H^n_0(0,1)\}$, where $n\in\N$, $\alpha\in H^n_0(0,1)$, $\alpha (x)>0$ in $(0,1).$ Under additional boundedness and integrability conditions on $\alpha$ with respect to $x^{2n} (1-x)^{2n},$ we prove that $(A_n,D(A_n))$ is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on $H^n_0(0,1)$. Analyticity results are also proved in $H^n (0,1).$ In particular, all results work well when $\alpha (x)=x^{j} (1-x)^{j}$ for $|j-n|<1/2$. Hardy type inequalities are also obtained.
keywords: analytic semigroups. degenerate operators Selfadjointness
Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains
Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Enrico Obrecht Silvia Romanelli
Communications on Pure & Applied Analysis 2016, 15(6): 2475-2487 doi: 10.3934/cpaa.2016045
We study nonsymmetric second order elliptic operators with Wentzell boundary conditions in general domains with sufficiently smooth boundary. The ambient space is a space of $L^p$- type, $1\le p\le \infty$. We prove the existence of analytic quasicontractive $(C_0)$-semigroups generated by the closures of such operators, for any $1< p< \infty$. Moreover, we extend a previous result concerning the continuous dependence of these semigroups on the coefficients of the boundary condition. We also specify precisely the domains of the generators explicitly in the case of bounded domains and $1 < p < \infty$, when all the ingredients of the problem, including the boundary of the domain, the coefficients, and the initial condition, are of class $C^{\infty}$.
keywords: Wentzell boundary conditions Nonsymmetric elliptic operators on general domains continuous dependence. perturbation of symmetric elliptic operators analytic semigroups
Gisèle Ruiz Goldstein Alain Miranville
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): i-ii doi: 10.3934/dcds.2013.33.11i
Jerome Arthur Goldstein (Jerry) was born on August 5, 1941 in Pittsburgh, PA. He attended Carnegie Mellon University, then called Carnegie Institute of Technology, where he earned his Bachelors of Science degree (1963), Masters of Science degree (1964) and Ph.D. (1967) in Mathematics. Jerry took a postdoctoral position at the Institute for Advanced Study in Princeton, followed by an Assistant Professorship at Tulane University. He became a Full Professor at Tulane University in 1975. After twenty-four years there, Jerry moved ``upriver" to join the faculty (and his wife, Gisèle) at Louisiana State University, where he was Professor of Mathematics from 1991 to 1996. In 1996 Jerry and Gisèle moved to the University of Memphis.

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Continuous dependence in hyperbolic problems with Wentzell boundary conditions
Giuseppe Maria Coclite Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Silvia Romanelli
Communications on Pure & Applied Analysis 2014, 13(1): 419-433 doi: 10.3934/cpaa.2014.13.419
Let $\Omega$ be a smooth bounded domain in $R^N$ and let \begin{eqnarray} Lu=\sum_{j,k=1}^N \partial_{x_j}\left(a_{jk}(x)\partial_{x_k} u\right), \end{eqnarray} in $\Omega$ and \begin{eqnarray} Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\partial_{\tau_k}\left(b_{jk}(x)\partial_{\tau_j}u\right)=0, \end{eqnarray} on $\partial\Omega$ define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary. Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space. If we have a sequence of such operators $S_0,S_1,S_2,...$ with corresponding coefficients \begin{eqnarray} \Phi_n=(a_{jk}^{(n)},b_{jk}^{(n)}, \beta_n,\gamma_n,q_n) \end{eqnarray} satisfying $\Phi_n\to\Phi_0$ uniformly as $n\to\infty$, then $u_n(t)\to u_0(t)$ where $u_n$ satisfies \begin{eqnarray} i\frac{du_n}{dt}=S_n^m u_n, \end{eqnarray} or \begin{eqnarray} \frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0, \end{eqnarray} or \begin{eqnarray} \frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0, \end{eqnarray} for $m=1,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$. This includes Schrödinger equations, damped and undamped wave equations, and telegraph equations.
keywords: Wentzell boundary conditions higher order boundary operators. continuous dependence Wave equation semigroup approximation
Gisèle Ruiz Goldstein Jerome A. Goldstein Alain Miranville
Discrete & Continuous Dynamical Systems - A 2008, 22(4): i-ii doi: 10.3934/dcds.2008.22.4i
This special issue consists of invited and carefully refereed papers on specific topics related to evolution equations, semigroup theory and related problems. Indeed, we thought that it would be very valuable to produce such a volume on important and active areas of research.
keywords: XXXX
Equipartition of energy for nonautonomous wave equations
Gisèle Ruiz Goldstein Jerome A. Goldstein Fabiana Travessini De Cezaro
Discrete & Continuous Dynamical Systems - S 2017, 10(1): 75-85 doi: 10.3934/dcdss.2017004
Consider wave equations of the form
$\begin{align*}u''(t)+ A^2u(t)=0\end{align*}$
with $A$ an injective selfadjoint operator on a complex Hilbert space
. The kinetic, potential, and total energies of a solution $u$ are
$\begin{align*}K(t)= \| u'(t)\|^2, P(t)= \|Au(t)\|^2, E(t) = K(t)+P(t).\end{align*}$
Finite energy solutions are those mild solutions for which
is finite. For such solutions
$E(t)= E(0)$
, that is, energy is conserved, and asymptotic equipartition of energy
$\begin{align*}\lim_{t \longrightarrow ± ∞}K(t) = \lim_{t \longrightarrow ± ∞}P(t) = \frac{E(0)}{2}\end{align*}$
holds for all finite energy mild solutions iff
$e^{itA}\longrightarrow 0$
in the weak operator topology. In this paper we present the first extension of this result to the case where
is time dependent.
keywords: Equipartition of energy nonautonomous system asymptotics wave equations
Kolmogorov equations perturbed by an inverse-square potential
Gisèle Ruiz Goldstein Jerome A. Goldstein Abdelaziz Rhandi
Discrete & Continuous Dynamical Systems - S 2011, 4(3): 623-630 doi: 10.3934/dcdss.2011.4.623
In this paper we present a nonexistence result of exponentially bounded positive solutions to a parabolic equation of Kolmogorov type with a more general drift term perturbed by an inverse square potential. This result generalizes the one obtained in [8]. Next we introduce some classes of nonlinear operators, related to the filtration operators and the $p$-Laplacian, and involving Kolmogorov operators. We establish the maximal monotonicity of some of these operators. In the third part we discuss the possibility of some nonexistence results in the context of singular potential perturbations of these nonlinear operators.
keywords: nonlinear parabolic equations Hardy's inequality positive solutions $p$-Kolmogorov operator. Inverse square potential Ornstein-Uhlenbeck operator critical constant
A convexified energy functional for the Fermi-Amaldi correction
Gisèle Ruiz Goldstein Jerome A. Goldstein Naima Naheed
Discrete & Continuous Dynamical Systems - A 2010, 28(1): 41-65 doi: 10.3934/dcds.2010.28.41
Consider the Thomas-Fermi energy functional $E$ for a spin polarized atom or molecule with $N_{1} $ [resp. $N_{2}$] spin up [resp. spin down] electrons and total positive molecular charge Z. Incorporating the Fermi-Amaldi correction as Benilan, Goldstein and Goldstein did, $E$ is not convex. By replacing $E$ by a well-motivated convex minorant $ \mathcal{E}$ ,we prove that $ \mathcal{E} $ has a unique minimizing density $( \rho _{1},\rho _{2}) \ $ when $N_{1}+N_{2}\leq Z+1\ $and $N_{2}\ $is close to $N_{1}.$
keywords: $L^{1} $constrained minimization ground state electron density Fermi-Amaldi correction convex minorant spin polarized system degree theory Thomas-Fermi theory

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