Optimal Hardy inequalities for general elliptic operators with improvements
Craig Cowan
Communications on Pure & Applied Analysis 2010, 9(1): 109-140 doi: 10.3934/cpaa.2010.9.109
We establish Hardy inequalities of the form

$ \int_\Omega | \nabla u|_A^2 dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2}u^2dx, \qquad u \in H_0^1(\Omega) \qquad\qquad (1)$

where $ E$ is a positive function defined in $ \Omega$, -div$(A \nabla E)$ is a nonnegative nonzero finite measure in $ \Omega$ which we denote by $ \mu$ and where $ A(x)$ is a $ n \times n$ symmetric, uniformly positive definite matrix defined in $ \Omega$ with $ | \xi |_A^2:= A(x) \xi \cdot \xi$ for $ \xi \in \mathbb{R}^n$. We show that (1) is optimal if $ E=0$ on $ \partial \Omega$ or $ E=\infty$ on the support of $ \mu$ and is not attained in either case. When $ E=0$ on $\partial \Omega$ we show

$ \int_\Omega | \nabla u|_A^2dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2}u^2dx + \frac{1}{2} \int_\Omega \frac{u^2}{E} d \mu, \qquad u \in H_0^1(\Omega)\qquad (2) $

is optimal and not attained. Optimal weighted versions of these inequalities are also established. Optimal analogous versions of (1) and (2) are established for $p$≠ 2 which, in the case that $ \mu$ is a Dirac mass, answers a best constant question posed by Adimurthi and Sekar (see [1]).
We examine improved versions of the above inequalities of the form

$\int_\Omega | \nabla u|_A^2dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2} u^2dx + \int_\Omega V(x) u^2dx, \qquad u \in H_0^1(\Omega).\qquad (3)$

Necessary and sufficient conditions on $V$ are obtained (in terms of the solvability of a linear pde) for (3) to hold. Analogous results involving improvements are obtained for the weighted versions.
In addition we obtain various results concerning the above inequalities valid for functions $ u$ which are nonzero on the boundary of $ \Omega$. We also examine the nonquadradic case ,ie. $p$ ≠2 of the above inequalities.

keywords: Hardy inequalities general elliptic operators.
Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications
Asadollah Aghajani Craig Cowan
Discrete & Continuous Dynamical Systems - A 2019, 39(5): 2731-2742 doi: 10.3934/dcds.2019114
In this paper we consider positive supersolutions of the nonlinear elliptic equation
$ - \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega, $
$ 0\le p<1 $
$ \Omega $
is an arbitrary domain (bounded or unbounded) in
$ {\mathbb{R}}^N $
$ N\ge 2 $
$ f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+} $
$ (0 < a_{f} \leq +\infty) $
is a non-decreasing continuous function and
$ \rho: \Omega \rightarrow \mathbb{R} $
is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions
$ u $
at each point
$ x\in\Omega $
$ \nabla u\not\equiv0 $
in a neighborhood of
$ x $
. As applications, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains
$ \Omega $
with the property that
$ \sup_{x\in\Omega}dist (x, \partial\Omega) = \infty $
. In particular when
$ \rho(x) = |x|^\beta $
$ \beta\in {\mathbb{R}} $
) and
$ f(u) = u^q $
$ q+p>1 $
then every positive supersolution in an exterior domain is eventually constant if
$ (N-2)q+p(N-1)< N+\beta. $
keywords: Nonlinear elliptic problems Liouville type theorems dead core supersolutions gradient term
Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains
Craig Cowan Pierpaolo Esposito Nassif Ghoussoub
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1033-1050 doi: 10.3934/dcds.2010.28.1033
We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $ \R^N$, with the Navier boundary condition $ u=\Delta u =0 $ on δΩ. We establish energy estimates which show that for any non-decreasing convex and superlinear nonlinearity $f$ with $f(0)=1$, the extremal solution u * is smooth provided $N\leq 5$. If in addition $\lim$i$nf_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}>0$, then u * is regular for $N\leq 7$, while if $\gamma$:$= \lim$s$up_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}<+\infty$, then the same holds for $N < \frac{8}{\gamma}$. It follows that u * is smooth if $f(t) = e^t$ and $ N \le 8$, or if $f(t) = (1+t)^p$ and $N< \frac{8p}{p-1}$. We also show that if $ f(t) = (1-t)^{-p}$, $p>1$ and $p\ne 3$, then u * is smooth for $N \leq \frac{8p}{p+1}$. While these results are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established on radial domains, e.g., u * is smooth for $ N \le 12$ when $ f(t) = e^t$ [11], and for $ N \le 8$ when $ f(t) = (1-t)^{-2}$ [9] (see also [22]).
keywords: Stable solution Superlinear and Singular nonlinearities. Extremal solution

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