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Christopher Grumiau

Publications and source records attributed to Christopher Grumiau.

7 recordsLinked to original sources
arXiv2019

Concordance probability in a big data setting: application in non-life insurance

The concordance probability or C-index is a popular measure to capture the discriminatory ability of a regression model. In this article, the definition of this measure is adapted to the specific needs of the frequency and severity model, typically used during the technical pricing of a non-life insurance product. Due to the typical large sample size of the frequency data in particular, two different adaptations of the estimation procedure of the concordance probability are presented. Note that the latter procedures can be applied to all different versions of the concordance probability.

math.AP↗
arXiv2016

Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem \[ \begin{cases} -Δu+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_νu=0,&\text{ on } \partial B_{R}, \end{cases} \] that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\to\infty$ (in particular, for supercritical exponents) or as $R \to \infty$ for any fixed value of $p>2$, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for $p$ and $R$ so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.

math.AP↗
arXiv2013

Lane Emden problems with large exponents and singular Liouville equations

We consider the Lane-Emden Dirichlet problem -Δu = \abs{u}^{p-1}u, in B, u =0, on \partial B, where $p>1$ and $B$ denotes the unit ball in $\IR^2$. We study the asymptotic behavior of the least energy nodal radial solution $u_p$, as $p\rightarrow +\infty$. Assuming w.l.o.g. that $u_p(0) < 0$, we prove that a suitable rescaling of the negative part $u_p^-$ converges to the unique regular solution of the Liouville equation in $\IR^2$, while a suitable rescaling of the positive part $u_p^+$ converges to a (singular) solution of a singular Liouville equation in $\IR^2$. We also get exact asymptotic values for the $L^\infty$-norms of $u_p^-$ and $u_p^+$, as well as an asymptotic estimate of the energy. Finally, we have that the nodal line $\N_p:={x\in B : \abs{x}= r_p}$ shrinks to a point and we compute the rate of convergence of $r_p$.

math.AP↗
arXiv2013

Convergence of a mountain pass type algorithm for strongly indefinite problems and systems

For a functional $\E$ and a peak selection that picks up a global maximum of $\E$ on varying cones, we study the convergence up to a subsequence to a critical point of the sequence generated by a mountain pass type algorithm. Moreover, by carefully choosing stepsizes, we establish the convergence of the whole sequence under a "localization" assumption on the critical point. We illustrate our results with two problems: an indefinite Schr\"odinger equation and a superlinear Schr\"odinger system.

math.AP↗
arXiv2012

Nonlinear Schrödinger problems: symmetries of some variational solutions

In this paper, we are interested in the nonlinear Schrödinger problem $-Δu + Vu = \abs{u}^{p-2}u$ submitted to the Dirichlet boundary conditions. We consider $p>2$ and we are working with an open bounded domain $Ω\subset\IR^N$ ($N\geq 2$). Potential $V$ satisfies $\max(V,0)\in L^{N/2}(Ω)$ and $\min(V,0)\in L^{+\infty}(Ω)$. Moreover, $-Δ+ V$ is positive definite and has one and only one principal eigenvalue. When $p\simeq 2$, we prove the uniqueness of the solution once we fix the projection on an eigenspace of $-Δ+ V$. It implies partial symmetries (or symmetry breaking) for ground state and least energy nodal solutions. In the litterature, the case $V\equiv 0$ has already been studied. Here, we generalize the technique at our case by pointing out and explaining differences. To finish, as illustration, we implement the (modified) mountain pass algorithm to work with $V$ negative, piecewise constant or not bounded. It permits us to exhibit direct examples where the solutions break down the symmetries of $V$.

math.AP↗
arXiv2012

Lane Emden problems: asymptotic behavior of low energy nodal solutions

We study the nodal solutions of the Lane Emden Dirichlet problem $-Δu = |u|^{p-1}u with DBC on a smooth bounded domain $Ω$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_Ω\abs{\nabla u_p}^2\to 16πe\quad\hbox{as}p\rightarrow+\infty\qquad (*)$ and we are interested in the shape and the asymptotic behavior as $p\rightarrow+\infty$. First we prove that (*) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that $pu_p$ can be characterized as the difference of two Green's functions and the nodal line intersects the boundary of $Ω$, for large $p$.

math.AP↗