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Pedro M. Girão

Publications and source records attributed to Pedro M. Girão.

At least 19 recordsLinked to original sources
arXiv2024

Regularity of a double null coordinate system for Kerr-Newman-de Sitter spacetimes

We construct a double null coordinate system $(u,v,θ_\star,ϕ_\star)$ for Kerr-Newman-de Sitter spacetimes and prove that the two-spheres given by the intersection of the hypersurfaces $u=\mbox{constant}$ and $v=\mbox{constant}$ are $C^\infty$ in Boyer-Lindquist coordinates (including at the "poles"). The null coordinates allow one to immediately extend some results previously proven for Kerr. As an example, we illustrate how Sbierski's result, for the wave equation on the black hole interior, for Reissner-Nordström and Kerr spacetimes, applies to Kerr-Newman-de Sitter spacetimes.

gr-qc↗
arXiv2019

Solutions of the wave equation bounded at the Big Bang

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_gϕ=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(Σ)$ (where $Σ=\mathbb{R}^n, \mathbb{S}^n$ or $\mathbb{H}^n$ models the spatial hypersurfaces) there exists a unique solution $ϕ$ of the wave equation converging to $A$ in $H^1(Σ)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(Σ)$.

gr-qc↗
arXiv2019

Higher Order Linear Stability and Instability of Reissner-Nordström's Cauchy Horizon

We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner-Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime, whose restrictions to the event horizon have compact support. We provide criteria, in terms of surface gravities, for the waves to remain in $C^l$, $l\geq 1$, up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in $C^1$ and $H^1$.

gr-qc↗
arXiv2018

On the occurrence of mass inflation for the Einstein-Maxwell-scalar field system with a cosmological constant and an exponential Price law

In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along the event horizon. More precisely, we construct open sets of characteristic data which, on the outgoing initial null hypersurface (taken to be the event horizon), converges exponentially to a reference Reissner-Nördstrom black hole at infinity. We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflation may or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with continuous metric and Christoffel symbols in $L^2_{\rm loc}$, thus violating the Christodoulou-Chruściel version of strong cosmic censorship.

gr-qc↗
arXiv2017

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 3: Mass inflation and extendibility of the solutions

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant $Λ$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold. In the first part of this series we established the well posedness of the characteristic problem, whereas in the second part we studied the stability of the radius function at the Cauchy horizon. In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when $Λ>0$.

gr-qc↗
arXiv2015

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 2: Structure of the solutions and stability of the Cauchy horizon

This paper is the second part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant $Λ$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold. In the first paper of this sequence, we established well posedness of the characteristic problem with general initial data. In this second paper, we generalize the results of Dafermos on the stability of the radius function at the Cauchy horizon by including a cosmological constant. This requires a considerable deviation from the strategy followed by Dafermos, focusing on the level sets of the radius function instead of the red-shift and blue-shift regions. We also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin. In the third and final paper, we will consider the issue of mass inflation and extendibility of solutions beyond the Cauchy horizon.

gr-qc↗
arXiv2014

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 1: Well posedness and breakdown criterion

This paper is the first part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant $Λ$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development (MGHD) as a "suitably regular" Lorentzian manifold. In this first part we establish well posedness of the Einstein equations for characteristic data satisfying the minimal regularity conditions leading to classical solutions. We also identify the appropriate notion of maximal solution, from which the construction of the corresponding MGHD follows, and determine breakdown criteria. This is the unavoidable starting point of the analysis; our main results will depend on the detailed understanding of these fundamentals. In the second part of this series we study the stability of the radius function at the Cauchy horizon. In the third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur; in fact, it is even possible to have (non-isometric) extensions of the spacetime across the Cauchy horizon as classical solutions of the Einstein equations.

gr-qc↗
arXiv2014

A family of sharp inequalities for Sobolev functions

Let $N\geq 5$, $Ω$ 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 $α_{0}(q,a,Ω)>0$ such that, for all $u\in H^1(Ω)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}{|u|_{2^*}^2}\leq||u||^2+α_{0} \left(\frac{||u||}{|u|_{2^*}^{2^*/2}}\right)^s|u|_{q}^{qt},\qquad{(I)_{q}}$$ where the norms are over $Ω$. Inequality $(I)_{2^\flat}$ is due to M. Zhu.

math.AP↗
arXiv2014

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 $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$, with $N\geq 5$, $a>0$, $α\geq 0$ 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 $$ \left\{\begin{array}{ll} -Δu+au=u^{2^*-1}-αu^{q-1}&\mbox{in}\ Ω,\\ u>0&\mbox{in}\ Ω,\\ \frac{\partial u}{\partialν}=0&\mbox{on}\ \partialΩ. \end{array}\right. $$ Namely, we prove that when $q=\frac{2(N-1)}{N-2}$ there exists an $α_{0}>0$ such that the problem has a least energy solution if $α<α_{0}$ and has no least energy solution if $α>α_{0}$.

math.AP↗
arXiv2014

A sharp inequality for Sobolev functions

Let $N\geq 5$, $a>0$, $Ω$ 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 $α_{0}>0$ such that, for all $u\in H^1(Ω)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}\leq\frac{||u||^2}{|u|_{2^*}^2}\left(1+α_{0}\frac{|u|_{2^\#}^{2^\#}}{||u||\cdot|u|_{2^*}^{2^*/2}}\right).$$ This inequality implies Cherrier's inequality.

math.AP↗
arXiv2014

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.

math.NA↗
arXiv2014

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.

math.NA↗
arXiv2014

Multibump nodal solutions for an indefinite nonhomogeneous elliptic problem

We construct multibump nodal solutions of the elliptic equation $$ -Δu=a^+[λu+ f(\, \cdot\,, u)]-μa^- g(\, \cdot\,, u) $$ in $H^1_0(Ω)$, when $μ$ is large, under appropriate assumptions, for $f$ superlinear and subcritical and such that the eigenvalues of the associated linearized operator on $H^1_0(\{x\inΩ:\: a(x)>0\})$ at zero, $u\longmapsto u-λ(-Δ)^{-1}(a^+ u)$, are positive. The solutions are of least energy in some Nehari-type set defined by imposing suitable conditions on orthogonal components of functions in $H^1_0(Ω)$.

math.AP↗
arXiv2014

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 $Ω\subset\mathbb{R}^N$, 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.

math.AP↗
arXiv2014

The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

We study extremal functions for a family of Poincaré-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 $p$.

math.AP↗
arXiv2014

On the Fučik 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 $λ$ 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 Fučik spectrum intersecting at $(λ,λ)$. 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 Fučik curves passing through $(λ,λ)\neq(0,0)$ and $p$ is a continuous function, sublinear at infinity.

math.AP↗