arXiv subjects

Francesca De Marchis

Publications and source records attributed to Francesca De Marchis.

At least 19 recordsLinked to original sources

arXiv2025

Stability and asymptotic behaviour of one-dimensional solutions in cylinders

We consider positive one-dimensional solutions of a Lane-Emden relative Dirichlet problem in a cylinder and study their stability/instability properties as the energy varies with respect to domain perturbations. This depends on the exponent $p >1$ of the nonlinearity and we obtain results for $p$ close to 1 and for $p$ large. This is achieved by a careful asymptotic analysis of the one-dimensional solution as $p \to 1$ or $p \to \infty$, which is of independent interest. It allows to detect the limit profile and other qualitative properties of these solutions.

math.AP↗

arXiv2024

Sharp boundary concentration for a two-dimensional nonlinear Neumann problem

We consider the elliptic equation $-Δu+ u=0$ in a bounded, smooth domain $Ω\subset\mathbb R^{2}$ subject to the nonlinear Neumann boundary condition $\partial u/\partialν= |u|^{p-1}u$ on $\partialΩ$ and study the asymptotic behavior as the exponent $p\rightarrow +\infty$ of families of positive solutions $u_p$ satisfying uniform energy bounds. We prove energy quantization and characterize the boundary concentration. In particular we describe the local asymptotic profile of the solutions around each concentration point and get sharp convergence results for the $L^{\infty}$-norm.

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arXiv2022

Critical points of the Moser-Trudinger functional on closed surfaces

Given a closed Riemann surface $(Σ,g)$ and any positive smooth weight, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional $$J_{p,β}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2β} \right)^{\frac{p}{2-p}}-\ln \int_Σ(e^{u_+^p}-1) f dv_g,$$ for every $p\in (1,2)$ and $β>0$, {or} for $p=1$ and $β\in (0,\infty)\setminus 4π\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):=\int_Σ(e^{u^2}-1)f dv_g$$ constrained to $\mathcal{E}_β:=\left\{v\text{ s.t. }\|v\|_{H^1}^2=β\right\}$ for any $β>0$.

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arXiv2021

Morse index computation for radial solutions of the {Hé}non problem in the disk

We compute the Morse index $\textsf{m}(u_{p})$ of any radial solution $u_{p}$ of the semilinear problem: \begin{equation} \label{problemaAbstract}\tag{P} \left\{ \begin{array}{lr} -Δu=|x|^α|u|^{p-1}u & \mbox{in } B\\ u=0 & \mbox{ on }\partial B \end{array} \right. \end{equation} where $B$ is the unit ball of $\mathbb R^{2}$ centered at the origin, $α\geq 0$ is fixed and $p>1$ is sufficiently large. In the case $α=0$, i.e. for the \emph{Lane-Emden problem}, this leads to the following Morse index formula \[\textsf{m}(u_{p}) = 4m^{2}-m-2, \] for $p$ large enough, where $m$ is the number of nodal domains of $u$.

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arXiv2018

Morse index and uniqueness of positive solutions of the Lane-Emden problem in planar domains

We compute the Morse index of $1$-spike solutions of the semilinear elliptic problem \begin{equation}\label{abstr} \tag{$\mathcal P_p$} \begin{cases} -Δu= u^p & \text{in $Ω$} \\ u=0 & \text{on $\partialΩ$} \\ u>0 & \text{in $Ω$.} \end{cases} \end{equation} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $p>1$ is sufficiently large. When $Ω$ is convex, our result, combined with the characterization in [22], a result in [41] and with recent uniform estimates in \cite{Sirakov}, gives the uniqueness of the solution to \eqref{abstr}, for $p$ large. This proves, in dimension two and for $p$ large, a conjecture by Gidas-Ni-Nirenberg [29].

math.AP↗

arXiv2018

Asymptotic analysis and energy quantization for the Lane-Emden problem in dimension two

We complete the study of the asymptotic behavior, as $p\rightarrow +\infty$, of the positive solutions to \[ \left\{\begin{array}{lr}-Δu= u^p & \mbox{in}Ω\\ u=0 &\mbox{on}\partial Ω\end{array}\right. \] when $Ω$ is any smooth bounded domain in $\mathbb R^2$, started in [4]. In particular we show quantization of the energy to multiples of $8πe$ and prove convergence to $\sqrt{e}$ of the $L^{\infty}$-norm, thus confirming the conjecture made in [4].

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arXiv2017

Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials

In this paper we consider a mean field problem on a compact surface with conical singularities. This problem appears in the Gaussian curvature prescription problem in Geometry, and also in the Electroweak Theory and in the abelian Chern-Simons-Higgs model in Physics. In this paper we focus on the case of sign-changing potentials, and we give results on compactness, existence and multiplicity of solutions.

math.AP↗

arXiv2017

Prescribed Gauss curvature problem on singular surfaces

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $Σ$ admitting conical singularities of orders $α_i$'s at points $p_i$'s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min-max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $χ(Σ)+\sum_i α_i$ approaches a positive even integer, where $χ(Σ)$ is the Euler characteristic of the surface $Σ$.

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arXiv2016

Existence of stationary turbulent flows with variable positive vortex intensity

We prove the existence of stationary turbulent flows with arbitrary positive vortex circulation on non simply connected domains. Our construction yields solutions for all real values of the inverse temperature with the exception of a quantized set, for which blow-up phenomena may occur. Our results complete the analysis initiated in [J. Diff. Equ. 260 (2016), 339-369].

math.AP↗

arXiv2016

Asymptotic profile of positive solutions of Lane-Emden problems in dimension two

We consider families $u_p$ of solutions to the problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-Δu= u^p & \mbox{ in }Ω\\ u>0 & \mbox{ in }Ω\\ u=0 & \mbox{ on }\partial Ω\end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $Ω$ is a smooth bounded domain of $\mathbb R^2$. We give a complete description of the asymptotic behavior of $u_p$ as $p\rightarrow +\infty$, under the condition \[p\int_Ω |\nabla u_p|^2\,dx\rightarrow β\in\mathbb R\qquad\mbox{ as $p\rightarrow +\infty$}.\]

math.AP↗

arXiv2016

A Morse index formula for radial solutions of Lane-Emden problems

We consider the semilinear Lane-Emden problem: \begin{equation}\label{problemAbstract}\left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq3$, centered at the origin and $1<p<p_S$, $p_S=\frac{N+2}{N-2}$. We prove that for any radial solution $u_p$ of \eqref{problemAbstract} with $m$ nodal domains its Morse index $\mathsf{m}(u_p)$ is given by the formula \[\mathsf{m}(u_p)=m+N(m-1)\] if $p$ is sufficiently close to $p_S$.

math.AP↗

arXiv2016

Asymptotic analysis for the Lane-Emden problem in dimension two

We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }Ωu=0\qquad\qquad\qquad\mbox{ on }\partial Ω\end{array}\right. \end{equation} when $p>1$ and $Ω\subset\mathbb R^2$ is a smooth bounded domain. The aim of the paper is to survey some recent results on the asymptotic behavior of solutions of (1) as the exponent $p\rightarrow \infty $.

math.AP↗

arXiv2016

Exact Morse index computation for nodal radial solutions of Lane-Emden problems

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq2$, centered at the origin and $1<p<p_S$, with $p_S=+\infty$ if $N=2$ and $p_S=\frac{N+2}{N-2}$ if $N\geq3$. Our main result is to prove that in dimension $N=2$ the Morse index of the least energy sign-changing radial solution $u_p$ of \eqref{problemAbstract} is exactly $12$ if $p$ is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in $\mathbb R^N$ in any dimension $N\geq2$.

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arXiv2014

Morse index and sign changing bubble towers for Lane-Emden problems

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract}\left\{ \begin{array}{lr} -Δu= |u|^{p-1}u\qquad \mbox{ in }Ω\\ u=0\qquad\qquad\qquad\mbox{ on }\partial Ω\end{array} \right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $Ω$ is a smooth bounded symmetric domain of $\mathbb R^2$. We show that for families $(u_p)$ of sign-changing symmetric solutions of \eqref{problemAbstract} an upper bound on their Morse index implies concentration of the positive and negative part, $u_p^\pm$, at the same point, as $p\to+\infty$. Then an asymptotic analysis of $u_p^+$ and $u_p^-$ shows that the asymptotic profile of $(u_p)$, as $p\to+\infty$, is that of a tower of two different bubbles.

math.AP↗

arXiv2014

Blow up of solutions of semilinear heat equations in non radial domains of $\mathbb R^2$

We consider the semilinear heat equation \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll}v_t-Δv= |v|^{p-1}v & \mbox{in}Ω\times (0,T)\\ v=0 & \mbox{on}\partial Ω\times (0,T)\\ v(0)=v_0 & \mbox{in}Ω\end{array}\right.\tag{$\mathcal P_p$} \end{equation} where $p>1$, $Ω$ is a smooth bounded domain of $\mathbb R^2$, $T\in (0,+\infty]$ and $v_0$ belongs to a suitable space. We give general conditions for a family $u_p$ of sign-changing stationary solutions of \eqref{problemAbstract}, under which the solution of \eqref{problemAbstract} with initial value $v_0=λu_p$ blows up in finite time if $|λ-1|>0$ is sufficiently small and $p$ is sufficiently large. Since for $λ=1$ the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin. In previous paper by Dickstein, Pacella and Sciunzi this phenomenon has already been observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric. Our conditions are more general and we provide examples of stationary solutions $u_p$ which are not radial and exhibit the same behavior.

math.AP↗

arXiv2014

Asymptotic analysis and sign changing bubble towers for Lane-Emden problems

We consider the semilinear Lane-Emden problem in a smooth bounded domain of the plane. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions as the exponent p of the nonlinearity goes to infinity. Among other results we show, under some symmetry assumptions on the domain, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p goes to infinity, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in the plane.

math.AP↗

arXiv2013

Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence

We are motivated by the study of the Microcanonical Variational Principle within the Onsager's description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and "thin" enough domains in the supercritical (with respect to the Moser-Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known \un{only} on multiply connected domains. Then we study the structure of these solutions by the analysis of their linearized problems and also obtain a new uniqueness result for solutions of the Mean Field Equation on thin domains whose energy is uniformly bounded from above. Finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and use it together with all the results obtained so far to solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is eventually shown to be concave.

math.AP↗

arXiv2013

Sign changing solutions of Lane Emden problems with interior nodal line and semilinear heat equations

We consider the semilinear Lane Emden problem in a smooth bounded simply connected domain in the plane, invariant by the action of a finite symmetry group G. We show that if the orbit of each point in the domain, under the action of the group G, has cardinality greater than or equal to four then, for p sufficiently large, there exists a sign changing solution of the problem with two nodal regions whose nodal line does not touch the boundary of the domain. This result is proved as a consequence of an analogous result for the associated parabolic problem.

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