Kodai Mathematical Journal

Conservation of the mass for solutions to a class of singular parabolic equations

Ahmad Z. Fino, Fatma Gamze Düzgün, Vincenzo Vespri

2014 Volume 37 Issue 3 Pages 519-531
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674606

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In this paper we deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L^∞ coefficients, whose prototypes are the p-Laplacian ($\frac{2N}{N+1}$ < p < 2) and the Porous medium equation ($(\frac{N-2}{N})_+$ < m < 1). In this range of the parameters p and m, we are in the so called fast diffusion case. We prove that the initial mass is preserved for all the times.

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Lower bounds for blow-up time in a parabolic problem with a gradient term under various boundary conditions

Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro

2014 Volume 37 Issue 3 Pages 532-543
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674607

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This paper deals with the blow-up phenomena of the solution u of a nonlinear parabolic problem with a gradient nonlinearity and time dependent coefficients. By using techniques based on Sobolev type and differential inequalities, we derive explicit lower bounds for the blow-up time, if blow-up occurs, when different boundary conditions are taken into account.

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Prices in the utility function and demand monotonicity

Emilio Barucci, Filippo Gazzola

2014 Volume 37 Issue 3 Pages 544-567
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674608

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We analyze utility functions when they depend both on the quantity of the goods consumed by the agent and on the prices of the goods. This approach allows us to model price effects on agents' preferences (e.g. the so-called Veblen effect and the Patinkin formulation). We provide sufficient conditions to observe demand monotonicity and substitution among goods. Power utility functions are investigated: we provide examples of price dependent utility functions that cannot be written as an increasing transformation of a classical utility function dependent only upon quantities.

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On the profile of solutions with time-dependent singularities for the heat equation

Toru Kan, Jin Takahashi

2014 Volume 37 Issue 3 Pages 568-585
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674609

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Let N ≥ 2, T ∈ (0,∞] and ξ ∈ C(0,T; R^N). Under some regularity condition for ξ, it is known that the heat equation
u_t − Δu = 0, x ∈ R^N \ {ξ(t)}, t ∈ (0,T)
has a solution behaving like the fundamental solution of the Laplace equation as x → ξ(t) for any fixed t. In this paper we construct a singular solution whose behavior near x = ξ(t) suddenly changes from that of the fundamental solution of the Laplace equation at some t.

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A note on Petty's theorem

Michele Marini, Guido de Philippis

2014 Volume 37 Issue 3 Pages 586-594
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674610

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In this short note we show how, by exploiting the regularity theory for solutions to the Monge-Ampère equation, Petty's equation characterizes ellipsoids without assuming any a priori regularity assumption.

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On the first Dirichlet Laplacian eigenvalue of regular polygons

Carlo Nitsch

2014 Volume 37 Issue 3 Pages 595-607
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674611

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The Faber-Krahn inequality in R² states that among all open bounded sets of given area the disk minimizes the first Dirichlet Laplacian eigenvalue. It was conjectured in [1] that for all N ≥ 3 the first Dirichlet Laplacian eigenvalue of the regular N-gon is greater than the one of the regular (N + 1)-gon of same area. This natural idea is suggested by the fact that the shape becomes more and more "rounded" as N increases and it is supported by clear numerical evidences. Aiming to settle such a conjecture, in this work we investigate possible ways to estimate the difference between eigenvalues of regular N-gons and (N + 1)-gons.

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Some remarks on a shape optimization problem

Francesco Della Pietra

2014 Volume 37 Issue 3 Pages 608-619
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674612

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Given a bounded open set Ω of Rⁿ, n ≥ 2, and α ∈ R, let us consider
μ(Ω,α) = $\min_{\substack{v\in W_{0}^{1,2}(\Omega)\\v\not\equiv 0}} \frac{\ds\int_{\Omega} |\nabla v|^{2}dx+\alpha \left|\ds\int_{\Omega}|v|v\,dx \right|}{\ds\int_{\Omega} |v|^{2}dx}$.
We study some properties of μ(Ω,α) and of its minimizers, and, depending on α, we determine the sets Ω_α among those of fixed measure such that μ(Ω_α,α) is the smallest possible.

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Eigenvalues of the Laplace-Beltrami operator on a large spherical cap under the Robin problem

Yoshitsugu Kabeya, Tatsuki Kawakami, Atsushi Kosaka, Hirokazu Ninomiya

2014 Volume 37 Issue 3 Pages 620-645
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674613

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Eigenvalues of the Laplace-Beltrami operator on a spherical cap is considered under the homogeneous Robin condition. The asymptotic behavior of eigenvalues and the influence of the eigenvalues by the boundary conditions are discussed as the cap becomes large so that the domain covers almost the whole sphere.

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Convergence rate in the weighted norm for a semilinear heat equation with supercritical nonlinearity

Yūki Naito

2014 Volume 37 Issue 3 Pages 646-667
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674614

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We study the behavior of solutions to the Cauchy problem for a semilinear heat equation with supercritical nonlinearity. It is known that two solutions approach each other if these initial data are close enough near the spatial infinity. In this paper, we give its sharp convergence rate in the weighted norms for a class of initial data. Proofs are given by a comparison method based on matched asymptotics expansion.

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A note on parabolic power concavity

Kazuhiro Ishige, Paolo Salani

2014 Volume 37 Issue 3 Pages 668-679
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674615

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We investigate parabolic power concavity properties of the solutions of the heat equation in Ω × [0,T), where Ω = Rⁿ or Ω is a bounded convex domain in Rⁿ.

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Interaction between fast diffusion and geometry of domain

Shigeru Sakaguchi

2014 Volume 37 Issue 3 Pages 680-701
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674616

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Let Ω be a domain in R^N, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider two fast diffusion equations ∂_tu = div(|∇u|^p-2∇u) and ∂_tu = Δu^m, where 1 < p < 2 and 0 < m < 1. Let u = u(x,t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value is a positive continuous function, or the Cauchy problem where the initial datum equals a nonnegative continuous function multiplied by the characteristic function of the set R^N\Ω. Choose an open ball B in Ω whose closure intersects ∂Ω only at one point, and let α > $\frac {(N+1)(2-p)}{2p}$ or α > $\frac {(N+1)(1-m)}{4}$. Then, we derive asymptotic estimates for the integral of u^α over B for short times in terms of principal curvatures of ∂Ω at the point, which tells us about the interaction between fast diffusion and geometry of domain.

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Local solvability of a fully nonlinear parabolic equation

Goro Akagi

2014 Volume 37 Issue 3 Pages 702-727
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674617

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This paper is concerned with the existence of local (in time) positive solutions to the Cauchy-Neumann problem in a smooth bounded domain of R^N for some fully nonlinear parabolic equation involving the positive part function r ∈ R $\mapsto$ (r)₊: = r ∨ 0. To show the local solvability, the equation is reformulated as a mixed form of two different sorts of doubly nonlinear evolution equations in order to apply an energy method. Some approximated problems are also introduced and the global (in time) solvability is proved for them with an aid of convex analysis, an energy method and some properties peculiar to the nonlinearity of the equation. Moreover, two types of comparison principles are also established, and based on these, the local existence and the finite time blow-up of positive solutions to the original equation are concluded as the main results of this paper.

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A note on Serrin's overdetermined problem

Giulio Ciraolo, Rolando Magnanini

2014 Volume 37 Issue 3 Pages 728-736
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674618

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We consider the solution of the torsion problem
−Δu = N in Ω, u = 0 on ∂Ω,
where Ω is a bounded domain in R^N.
Serrin's celebrated symmetry theorem states that, if the normal derivative u_ν is constant on ∂Ω, then Ω must be a ball. In [6], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate
r_e − r_i ≤ C_t(max_{Γ_t} u − min_{Γ_t} u)
for some constant C_t depending on t, where r_e and r_i are the radii of an annulus containing ∂Ω and Γ_t is a surface parallel to ∂Ω at distance t and sufficiently close to ∂Ω secondly, if in addition u_ν is constant on ∂Ω, show that
max_{Γ_t} u − min_{Γ_t} u = o(C_t) as t → 0⁺.
The estimate constructed in [6] is not sharp enough to achieve this goal. In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains Ω are ellipses.

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A decomposition theorem of the Möbius energy I: Decomposition and Möbius invariance

Aya Ishizeki, Takeyuki Nagasawa

2014 Volume 37 Issue 3 Pages 737-754
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674619

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The Möbius energy, defined for closed curves embedded in Rⁿ, is decomposed into three parts. It is called the Möbius energy since it is invariant under Möbius transformations. An alternative proof of the Möbius invariance is given using the decomposition. Furthermore the invariance of each part of decomposition is discussed.

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Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions

Futoshi Takahashi

2014 Volume 37 Issue 3 Pages 755-768
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674620

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We consider a semilinear elliptic problem with the boundary reaction:
−Δu = 0 in Ω, $\frac{\partial u}{\partial \nu}$ + u = a(x) u^p + f(x) on ∂Ω,
where Ω ⊂ R^N, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, f ∈ L¹(∂Ω) are nonnegative functions, not identically equal to zero. We provide a necessary condition and a sufficient condition for the existence of positive very weak solutions of the problem. As a corollary, under some assumption of the potential function a, we prove that the problem has no positive solution for any nonnegative external force f ∈ L^∞(∂Ω), f $\not\equiv$ 0, even in the very weak sense.

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Convexity properties of Dirichlet integrals and Picone-type inequalities

Lorenzo Brasco, Giovanni Franzina

2014 Volume 37 Issue 3 Pages 769-799
Published: 2014
Released on J-STAGE: November 05, 2014

DOIhttps://doi.org/10.2996/kmj/1414674621

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We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

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