DCDS-S
Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces
Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Silvia Romanelli
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
CPAA
Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains
Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Enrico Obrecht Silvia Romanelli
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
CPAA
Continuous dependence in hyperbolic problems with Wentzell boundary conditions
Giuseppe Maria Coclite Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Silvia Romanelli
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
DCDS
Preface
Gisèle Ruiz Goldstein Jerome A. Goldstein Alain Miranville
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
DCDS-S
Kolmogorov equations perturbed by an inverse-square potential
Gisèle Ruiz Goldstein Jerome A. Goldstein Abdelaziz Rhandi
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
DCDS-B
Generators of Feller semigroups with coefficients depending on parameters and optimal estimators
Jerome A. Goldstein Rosa Maria Mininni Silvia Romanelli
We consider the realization of the operator $L_{\theta, a}u(x) $:$= x^{2 a}u''(x) \ + \ (a x^{2 a - 1} + \theta x^a)u'(x)$, acting on $C[0,+\infty]$, for $\theta\in\R$, $a\in\R$. We show that $L_{\theta, a}$, with the so called Wentzell boundary conditions, generates a Feller semigroup for any $\theta\in\R$, $a\in\R$. The problem of finding optimal estimators for the corresponding diffusion processes is also discussed, in connection with some models in financial mathematics. Here $C[0,+\infty]$ is the space of all real valued continuous functions on $[0,+\infty)$ which admit finite limit at $+\infty$.
keywords: stochastic differential equations optimal estimators. diffusion processes Feller semigroups operator semigroups
DCDS
A convexified energy functional for the Fermi-Amaldi correction
Gisèle Ruiz Goldstein Jerome A. Goldstein Naima Naheed
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
DCDS
A unified approach to weighted Hardy type inequalities on Carnot groups
Jerome A. Goldstein Ismail Kombe Abdullah Yener
We find a simple sufficient criterion on a pair of nonnegative weight functions
$V(x)$
and
$W(x) $
on a Carnot group
$\mathbb{G},$
so that the general weighted
$L^{p}$
Hardy type inequality
$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$
is valid for any
$φ ∈ C_{0}^{∞ }(\mathbb{G})$
and
$p>1.$
It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on
$\mathbb{G}.$
We also present some new results on two-weight
$L^{p}$
Hardy type inequalities with remainder terms on a bounded domain
$Ω$
in
$\mathbb{G}$
via a differential inequality.
keywords: Carnot groups weighted Hardy inequality Heisenberg-Pauli-Weyl inequality two-weight Hardy inequality
DCDS-S
Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations
Raluca Clendenen Gisèle Ruiz Goldstein Jerome A. Goldstein
In the dynamic or Wentzell boundary condition for elliptic, parabolic and hyperbolic partial differential equations, the positive flux coefficient $% \beta $ determines the weighted surface measure $dS/\beta $ on the boundary of the given spatial domain, in the appropriate Hilbert space that makes the generator for the problem selfadjoint. Usually, $\beta $ is continuous and bounded away from both zero and infinity, and thus $L^{2}\left( \partial \Omega ,dS\right) $ and $L^{2}\left( \partial \Omega ,dS/\beta \right) $ are equal as sets. In this paper this restriction is eliminated, so that both zero and infinity are allowed to be limiting values for $\beta $. An application includes the parabolic asymptotics for the Wentzell telegraph equation and strongly damped Wentzell wave equation with general $\beta $.
keywords: parabolic asymptotics degenerate flux. Dynamic boundary conditions telegraph equation Wentzell boundary conditions

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