A sharp inequality for Sobolev functions

Let $N\geq 5$, $a>0$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $2^*=\frac{2N}{N-2}$, $2^\#=\frac{2(N-1)}{N-2}$ and $||u||^2=|\nabla u|_{2}^2+a|u|_{2}^2$. We prove there exists an $\alpha_{0}>0$ such that, for all $u\in H^1(\Omega)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}\leq\frac{||u||^2}{|u|_{2^*}^2}\left(1+\alpha_{0}\frac{|u|_{2^\#}^{2^\#}}{||u||\cdot|u|_{2^*}^{2^*/2}}\right).$$ This inequality implies Cherrier's inequality.Comment: 4 page

A family of sharp inequalities for Sobolev functions

Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, ${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left. \int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}), \nabla u\in L^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}}|u|^{2^*}=1 \right\}$ and $||u||^2=|\nabla u|_{2}^2+a|u|_{2}^2$. We define ${2^\flat}= \frac{2N}{N-1}$, ${2^\#}=\frac{2(N-1)}{N-2}$ and consider $q$ such that ${2^\flat}\leq q\leq{2^\#}$. We also define $s=2-N+\frac{q}{{2^*}-q}$ and $t=\frac{2}{N-2}\cdot \frac{1}{{2^*}-q}$. We prove that there exists an $\alpha_{0}(q,a,\Omega)>0$ such that, for all $u\in H^1(\Omega)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}{|u|_{{2^*}}^2}\leq||u||^2+\alpha_{0} \left(\frac{||u||}{|u|_{{2^*}}^{2^*/2}}\right)^s|u|_{q}^{qt},\qquad{(I)_{q}}$$ where the norms are over $\Omega$. Inequality $(I)_{{2^\flat}}$ is due to M. Zhu.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1407.623

Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue

We construct the global bifurcation curves, solutions versus level of harvesting, for the steady states of a diffusive logistic equation on a bounded domain, under Dirichlet boundary conditions and other appropriate hypotheses, when $a$, the linear growth rate of the population, is below $\lambda_2+\delta$. Here $\lambda_2$ is the second eigenvalue of the Dirichlet Laplacian on the domain and $\delta>0$. Such curves have been obtained before, but only for $a$ in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for $a\leq\lambda_2$ and new information on the number of solutions for $a>\lambda_2$.Comment: This is an extended version of the published pape

Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature

Motion by weighted mean curvature is a geometric evolution law for surfaces and represents steepest descent with respect to anisotropic surface energy. It has been proposed that this motion could be computed numerically by using a "crystalline" approximation to the surface energy in the evolution law. In this paper we prove the convergence of this numerical method for the case of simple closed convex curves in the plane.Comment: 14 pages, 3 figure

On the Fu\v{c}ik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equation $$u_{tt}-u_{ss}=au^+-bu^-+p(s,t,u),$$ with periodic conditions in $t$ and homogeneous Dirichlet conditions in $s$, under the assumption that the ratio of the period to the length of the interval is two. When $p\equiv 0$ and $\lambda$ is a nonzero eigenvalue of the wave operator, we give a proof of the existence of two families of curves (which may coincide) in the Fu\v{c}ik spectrum intersecting at $(\lambda,\lambda)$. This result is known for some classes of self-adjoint operators (which does not cover the situation we consider here), but in a smaller region than ours. Our approach is based on a dual variational formulation and is also applicable to other operators, such as the Laplacian. In addition, we prove an existence result for the nonhomogeneous situation, when the pair $(a,b)$ is not `between' the Fu\v{c}ik curves passing through $(\lambda,\lambda)\neq(0,0)$ and $p$ is a continuous function, sublinear at infinity

The shape of extremal functions for Poincar\'e-Sobolev-type inequalities in a ball

We study extremal functions for a family of Poincar\'e-Sobolev-type inequalities. These functions minimize, for subcritical or critical $p\geq 2$, the quotient ${\|\nabla u\|_2}/{\|u\|_p}$ among all $u \in H^1(B)\setminus\{0\}$ with $\int_{B}u=0$. Here $B$ is the unit ball in $\mathbb{R}^N$. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of $B$. We also prove that, for $p$ close to $2$, minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis, and that, once the symmetry axis is fixed, they are unique (up to multiplication by a constant). In space dimension two, we prove that minimizers are not antisymmetric for large $p$

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in $\mathbb{R}^N$ and in a bounded domain $\Omega\subset\mathbb{R}^N$, with $N\geq 3$, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem

Sign changing solutions for elliptic equations with critical growth in cylinder type domains

We prove the existence of positive and of nodal solutions for $-\Delta u = |u|^{p-2}u+\mu |u|^{q-2}u$, $u\in {\rm H_0^1}(\Omega)$, where $\mu >0$ and $2<q<p=2N(N-2)$, for a class of open subsets $\Omega$ of $\mathbb{R}^N$ lying between two infinite cylinders.Comment: 13 page

Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature

Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a "crystalline" approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension.Comment: 28 pages, 4 figure

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, with $N\geq 5$, $a>0$, $\alpha\geq 0$ and $2^*=\frac{2N}{N-2}$. We show that the the exponent $q=\frac{2(N-1)}{N-2}$ plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem \left\{\begin{array}{ll} -\Delta u+au=u^{2^*-1}-\alpha u^{q-1}&\mbox{in}\ \Omega,\\ u>0&\mbox{in}\ \Omega,\\ \frac{\partial u}{\partial\nu}=0&\mbox{on}\ \partial\Omega. \end{array}\right. Namely, we prove that when $q=\frac{2(N-1)}{N-2}$ there exists an $\alpha_{0}>0$ such that the problem has a least energy solution if $\alpha<\alpha_{0}$ and has no least energy solution if $\alpha>\alpha_{0}$.Comment: 30 page