arXiv subjects

Rolando Magnanini

Publications and source records attributed to Rolando Magnanini.

At least 19 recordsLinked to original sources

arXiv2026

Short-time behavior and invariant surfaces for caloric functions with non-constant boundary values

We consider a Cauchy-Dirichlet problem for a heat equation with variable coefficients in non-divergence form. The initial values are assumed to be homogeneous, while the Dirichlet boundary values are non-constant. In this setting, we derive an asymptotic formula for the short-time behavior of the solution, which extends the celebrated Varadhan formula. We stress the fact that the boundary values are allowed to vanish or change sign. The formula is obtained by combining the original ideas of Varadhan with those of Evans and Ishii, pertaining to the theory of viscosity solutions, and some further remarks. In passing, we prove the elliptic counterpart of the formula. This concerns the slow-diffusion behavior of the solutions of the resolvent equation. Furthermore, for the case of the classical Laplace operator, we prove asymptotic formulas for the heat content and the mean value of the solution of the resolvent equation on spheres touching the boundary of the domain. These extend to the case of non-constant boundary values, certain formulas previously obtained by the second author and S. Sakaguchi. The formulas involve the principal curvatures at the touching point and the Dirichlet boundary values. We then use our formulas to describe time-invariant surfaces for the heat equation, in presence of non-constant Dirichlet boundary values. We give two symmetry results in case of non-negative Dirichlet boundary values and a non-existence result, when those values change sign.

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arXiv2026

Critical points of solutions of elliptic equations in divergence form in planar non simply connected domains with smooth or nonsmooth boundary

We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on its normal derivatives) along the boundary. The proof combines the argument principle introduced in [1] for elliptic equations with the representation formula (using quasi-conformal mappings) for operators in divergence form in simply connected domains [2]. The case of a degenerate coefficient is also treated where we combine the level lines technique and the maximum principle with the argument principle. Finally, some numerical experiments on illustrative examples are presented. [1] G. Alessandrini and R. Magnanini. The index of isolated critical points and solutions of elliptic equations in the plane. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19(4):567-589, 1992 [2] G. Alessandrini and R. Magnanini. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal., 25(5):1259-1268, 1994

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arXiv2024

A general integral identity with applications to a reverse Serrin problem

We prove a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that the first and third authors previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem. As an application, we prove a quantitative symmetry result for the reverse Serrin problem, which we introduce for the first time in this paper. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition.

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arXiv2024

Nonlinear asymptotic mean value characterizations of holomorphic functions

Starting from a characterization of holomorphic functions in terms of a suitable mean value property, we build some nonlinear asymptotic characterizations for complex-valued solutions of certain nonlinear systems, which have to do with the classical Cauchy-Riemann equations. From these asymptotic characterizations, we derive suitable asymptotic mean value properties, which are used to construct appropriate vectorial dynamical programming principles. The aim is to construct approximation schemes for the so-called contact solutions, recently introduced by N. Katzourakis, of the nonlinear systems here considered.

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arXiv2023

Quantitative symmetry in a mixed Serrin-type problem for a constrained torsional rigidity

We consider a mixed boundary value problem in a domain $Ω$ contained in a half-ball $B_+$ and having a portion $\bar{T}$ of its boundary in common with the curved part of $\partial B_+$. The problem has to do with some sort of constrained torsional rigidity. In this situation, the relevant solution $u$ satisfies a Steklov condition on $T$ and a homogeneous Dirichlet condition on $Σ= \partialΩ\setminus \bar{T} \subset B_+$. We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution in $Ω$ to its normal derivative $u_ν$ on $Σ$. A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for $u_ν$ on $Σ$: in fact, it turns out that $Σ$ must be a spherical cap that meets $T$ orthogonally. This result returns the one obtained by J. Guo and C. Xia under the stronger pointwise condition that the values of $u_ν$ be constant on $Σ$. A second important consequence is a set of stability bounds, which quantitatively measure how $Σ$ is far uniformly from being a spherical cap, if $u_ν$ deviates from a constant in the norm $L^1(Σ)$.

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arXiv2023

The interior Backus problem: local resolution in Hölder spaces

We prove an existence result for the Backus interior problem in the Euclidean ball. The problem consists in determining a harmonic function in the ball from the knowledge of the modulus of its gradient on the boundary. The problem is severely nonlinear. From a physical point of view, the problem can be interpreted as the determination of the velocity potential of an incompressible and irrotational fluid inside the ball from measurements of the velocity field's modulus on the boundary. The linearized problem is an irregular oblique derivative problem, for which a phenomenon of loss of derivatives occurs. As a consequence, a solution by linearization of the Backus problem becomes problematic. Here, we linearize the problem around the vertical height solution and show that the loss of derivatives does not occur for solutions which are either (vertically) axially symmetric or oddly symmetric in the vertical direction. A standard fixed point argument is then feasible, based on ad hoc weighted estimates in Hölder spaces.

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arXiv2021

Explicit complex-valued solutions of the 2D eikonal equation

We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of Helmholtz equation by the WKBJ or EWT methods. We obtain the complex-valued solutions (called eikonals) as parameterizations in a complex variable. We consider both the cases of constant and non-constant index of refraction. In both cases, the relevant parameterizations depend on some holomorphic function. In the case of non-constant index of refraction, the parametrization also depends on some extra exponential complex-valued function and on a quasi-conformal homeomorphism. This is due to the use of the theory of pseudo-analytic functions and the related similarity principle. The parameterizations give information about the formation of caustics and the light and shadow regions for the relevant eikonals.

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arXiv2021

Interpolating estimates with applications to some quantitative symmetry results

We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $L^p$ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.

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arXiv2021

Backus problem in geophysics: a resolution near the dipole in fractional Sobolev spaces

We consider Backus's problem in geophysics. This consists in reconstructing a harmonic potential outside the Earth when the intensity of the related field is measured on the Earth's surface. Thus, the boundary condition is (severely) nonlinear. The gravitational case is quite understood. It consists in the local resolution near a monopole, i.e. the potential generated by a point mass. In this paper, we consider the geomagnetic case. This consists in linearizing the field's intensity near the so-called dipole, a harmonic function which models the solenoidal potential of a magnet. The problem is quite difficult, because the resolving operator related to the linearized problem is generally unbounded. Indeed, existence results for Backus's problem in this framework are not present in the literature. In this work, we locally solve the geomagnetic version of Backus's problem in the axially symmetric case. In mathematical terms, we show the existence of harmonic functions in the exterior of a sphere, with given (boundary) field's intensity sufficiently close to that of a dipole and which have the same axial symmetry of a dipole. We also show that unique solutions can be selected by prescribing the average of the potential on the equatorial circle of the sphere. We obtain those solutions as series of spherical harmonics. The functional framework entails the use of fractional Sobolev Hilbert spaces on the sphere, endowed with a spectral norm. A crucial ingredient is the algebra structure of suitable subspaces.

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arXiv2021

An interpolating inequality for solutions of uniformly elliptic equations

We extend an inequality for harmonic functions, obtained in previous research by the authors, to the case of solutions of uniformly elliptic equations in divergence form, with merely measurable coefficients. The inequality for harmonic functions turned out to be a crucial ingredient in the study of the stability of the radial symmetry for Alexandrov's Soap Bubble Theorem and Serrin's problem. The proof of our inequality is based on a mean value property for elliptic operators by L. Caffarelli, proved in full detail by I. Blank and Z. Hao.

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arXiv2021

Variational $p$-harmonious functions: existence and convergence to $p$-harmonic functions

In a recent paper, the last three authors showed that a game-theoretic $p$-harmonic function $v$ is characterized by an asymptotic mean value property with respect to a kind of mean value $ν_p^r[v](x)$ defined variationally on balls $B_r(x)$. In this paper, in a domain $\Om\subset\RR^N$, $N\ge 2$, we consider the operator $μ_p^\ve$, acting on continuous functions on $\ol{\Om}$, defined by the formula $μ_p^\ve[v](x)=ν^{r_\ve(x)}_p[v](x)$, where $r_\ve(x)=\min[\ve,\dist(x,\Ga)]$ and $\Ga$ denotes the boundary of $Ω$. We first derive various properties of $μ^\ve_p$ such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function $u^\ve\in C(\ol{\Om})$ satisfying the Dirichlet-type problem: $$ u(x)=μ_p^\ve[u](x) \ \mbox{ for every } \ x\in\Om,\quad u=g \ \mbox{ on } \ \Ga, $$ for any given function $g\in C(\Ga)$. This result holds, if we assume the existence of a suitable notion of barrier for all points in $\Ga$. That $u^\ve$ is what we call the \textit{variational} $p$-harmonious function with Dirichlet boundary data $g$, and is obtained by means of a Perron-type method based on a comparison principle. \par We then show that the family $\{ u^\ve\}_{\ve>0}$ gives an approximation scheme for the viscosity solution $u\in C(\ol{\Om})$ of $$ \De_p^G u=0 \ \mbox{ in }\Om, \quad u=g \ \mbox{ on } \ \Ga, $$ where $\De_p^G$ is the so-called game-theoretic (or homogeneous) $p$-Laplace operator. In fact, we prove that $u^\ve$ converges to $u$, uniformly on $\ol{\Om}$ as $\ve\to 0$.

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arXiv2020

The location of hot spots and other extremal points

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we first consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with a mean convex boundary and compare that distance to the inradius of the relevant domain. For the torsional rigidity function, the obtained bound only depends on the space dimension. The more general case of a boundary which is not mean convex is also considered. However, the estimates also depend on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary. Also in the case of the first mode, the relevant bound only depends on the space dimension. Moreover, it largely improves on an earlier estimate obtained by the first author and co-authors, for convex domains. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation. The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semilinear equations.

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arXiv2020

Small diffusion and short-time asymptotics for Pucci operators

This paper presents asymptotic formulas in the case of the following two problems for the {\it Pucci's extremal operators} $\mathcal{M}^\pm$. It is considered the solution $u^\varepsilon(x)$ of $-\varepsilon^2 \mathcal{M}^\pm\left(\nabla ^2 u^\varepsilon\right)+u^\varepsilon=0$ in $Ω$ such that $u^\varepsilon=1$ on $Γ$. Here, $Ω\subset \mathbb{R}^N$ is a domain (not necessarily bounded) and $Γ$ is its boundary. It is also considered $v(x,t)$ the solution of $v_t - \mathcal{M}^\pm\left(\nabla^2 v\right)=0$ in $Ω\times (0,\infty)$, $v=1$ on $Γ\times(0,\infty)$ and $v=0$ on $Ω\times \{0\}$. In the spirit of their previous works, the authors establish the profiles as $\varepsilon$ or $t\to 0^+$ of the values of $u^\varepsilon(x)$ and $v(x,t)$ as well as of those of their $q$-means on balls touching $Γ$. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime.

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arXiv2019

Nearly optimal stability for Serrin's problem and the Soap Bubble theorem

We present new quantitative estimates for the radially symmetric configuration concerning Serrin's overdetermined problem for the torsional rigidity, Alexandrov's Soap Bubble Theorem, and other related problems. The new estimates improve on those obtained in two our previous articles, and are in some cases optimal.

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arXiv2019

Two-phase heat conductors with a surface of the constant flow property

We consider a two-phase heat conductor in $\mathbb R^N$ with $N \geq 2$ consisting of a core and a shell with different constant conductivities. We study the role played by radial symmetry for overdetermined problems of elliptic and parabolic type. First of all, with the aid of the implicit function theorem, we give a counterexample to radial symmetry for some two-phase elliptic overdetermined boundary value problems of Serrin-type. Afterwards, we consider the following setting for a two-phase parabolic overdetermined problem. We suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. A hypersurface in the domain has the constant flow property if at every of its points the heat flux across surface only depends on time. It is shown that the structure of the conductor must be spherical, if either there is a surface of the constant flow property in the shell near the boundary or a connected component of the boundary of the heat conductor is a surface of the constant flow property. Also, by assuming that the medium outside the conductor has a possibly different conductivity, we consider a Cauchy problem in which the conductor has initial inside temperature $0$ and outside temperature $1$. We then show that a quite similar symmetry result holds true.

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arXiv2018

Short time behaviour for game-theoretic $p$-caloric functions

We consider the solution of $u_t-Δ^G_p u=0$ in a (not necessarily bounded) domain, satisfying $u=0$ initially and $u=1$ on the boundary at all times. Here, $Δ^G_p u$ is the game-theoretic or normalized $p$-laplacian. We derive new precise asymptotic formulas for short times, that generalize the work of S. R. S. Varadhan for large deviations and that of the second author and S. Sakaguchi for the heat content of a ball touching the boundary. We also compute the short-time behavior of the $q$-mean of $u$ on such a ball. Applications to time-invariant level surfaces of $u$ are then derived.

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arXiv2018

Asymptotics for the resolvent equation associated to the game-theoretic $p$-laplacian

We consider the (viscosity) solution $u^\varepsilon$ of the elliptic equation $\varepsilon^2Δ_p^G u= u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, $Δ_p^G$ is the {\it game-theoretic or normalized $p$-laplacian}. We derive asymptotic formulas for $\varepsilon\to 0^+$ involving the values of $u^\varepsilon$, in the spirit of Varadhan's work \cite{Va}, and its $q$-mean on balls touching the boundary, thus generalizing that obtained in \cite{MS-AM} for $p=q=2$. As in a related parabolic problem, investigated in \cite{BM}, we link the relevant asymptotic behavior to the geometry of the domain.

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arXiv2017

Alexandrov, Serrin, Weinberger, Reilly: simmetry and stability by integral identities

The distinguished names in the title have to do with influential proofs of the celebrated Soap Bubble Theorem and of radial symmetry in certain overdetermined boundary value problems. We shall give an overeview of those results and indicate some of their ramifications. We will also show how more recent proofs uncover the path to some stability results for the relevant problems.

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