DCDS
On a generalized Wirtinger inequality
Gisella Croce Bernard Dacorogna
Discrete & Continuous Dynamical Systems - A 2003, 9(5): 1329-1341 doi: 10.3934/dcds.2003.9.1329
Let

$\alpha( p,q,r) =$inf{$\frac{|| u'||_p}{||u||_q}:u\in W_{p e r}^{1,p}( -1,1) $\{$ 0$}, $\int_{-1}^1|u|^{r-2} u=0$} .

We show that

$\alpha( p,q,r )=\alpha ( p,q,q)$ if $q\leq rp+r-1$

$\alpha( p,q,r) <\alpha( p,q,q) $ if $q> ( 2r-1) p$

generalizing results of Dacorogna-Gangbo-Subía and others.

keywords: direct methods in the calculus of variations best constant in Sobolev spaces special functions. Nonlinear Wirtinger inequality
DCDS-S
An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term
Gisella Croce
Discrete & Continuous Dynamical Systems - S 2012, 5(3): 507-530 doi: 10.3934/dcdss.2012.5.507
In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. The model problem is $$ \begin{equation} \left\{\begin{array}{11} -div\left(\frac{\nabla u}{(1+|u|)^p}\right) + \frac{|\nabla u|^{2}}{|u|^{\theta}} = f & \mbox{in $\Omega$,} \\ u = 0 & \mbox{on $\partial\Omega$,} \end{array} \right. \end{equation} $$ where $\Omega$ is an open bounded set of $\mathbb{R}^N$, $N\geq 3$ and $p, \theta>0$. The source $f$ is a positive function belonging to some Lebesgue space. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions, when $p>\theta-1$ and $\theta<2$.
keywords: Dirichlet condition Nonlinear elliptic problems distributional solution singular lower order term quadratic growth. degenerate coercivity
DCDS
$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem
Gisella Croce Nikos Katzourakis Giovanni Pisante
Discrete & Continuous Dynamical Systems - A 2017, 37(12): 6165-6181 doi: 10.3934/dcds.2017266
For
$\mathrm{H}∈ C^2(\mathbb{R}^{N\times n})$
and
$u :Ω \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$
, consider the system
$ \label{1}\mathrm{A}_∞ u\, :=\,\Big(\mathrm{H}_P \otimes \mathrm{H}_P + \mathrm{H}[\mathrm{H}_P]^\bot \mathrm{H}_{PP}\Big)(\text{D} u):\text{D}^2u\, =\,0. \tag{1}$
We construct
$\mathcal{D}$
-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our
$\mathcal{D}$
-solutions are
$W^{1,∞}$
-submersions and are obtained without any convexity hypotheses for
$\mathrm{H}$
, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions
$n≠ N$
.
keywords: Vectorial Calculus of Variations in L generalised solutions fully nonlinear systems ∞-Laplacian young measures singular value problem Baire Category method convex integration

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