1.
Convexity properties of Dirichlet integrals and Picone-type inequalities - Brasco, Lorenzo; Franzina, Giovanni
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.

2.
Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions - Takahashi, Futoshi
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 $subset$ R^{N}, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, f $\in$ L^{1}(∂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...

3.
A decomposition theorem of the Möbius energy I: Decomposition and Möbius invariance - Ishizeki, Aya; Nagasawa, Takeyuki
The Möbius energy, defined for closed curves embedded in R^{n}, 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.

4.
A note on Serrin's overdetermined problem - Ciraolo, Giulio; Magnanini, Rolando
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} ≤ $C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$ ¶ for some constant C_{t} depending on t, where r_{e} and r_{i}...

5.
Local solvability of a fully nonlinear parabolic equation - Akagi, Goro
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 $\in$ 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...

6.
Interaction between fast diffusion and geometry of domain - Sakaguchi, Shigeru
Let Ω be a domain in R^{N}, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider two fast diffusion equations ∂_{t}u = div(|∇u|^{p-2}∇u) and ∂_{t}u = 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...

7.
A note on parabolic power concavity - Ishige, Kazuhiro; Salani, Paolo
We investigate parabolic power concavity properties of the solutions of the heat equation in Ω × [0,T), where Ω = R^{n} or Ω is a bounded convex domain in R^{n}.

8.
Convergence rate in the weighted norm for a semilinear heat equation with supercritical nonlinearity - Naito, Yūki
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.

9.
Eigenvalues of the Laplace-Beltrami operator on a large spherical cap under the Robin problem - Kabeya, Yoshitsugu; Kawakami, Tatsuki; Kosaka, Atsushi; Ninomiya, Hirokazu
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.

10.
Some remarks on a shape optimization problem - Della Pietra, Francesco
Given a bounded open set Ω of R^{n}, n ≥ 2, and α $\in$ R, let us consider ¶
$\mu(\Omega,\alpha)=\min_{\substack{v\in W_{0}^{1,2}(\Omega)\\v\not\equiv 0}} \frac{\ds\int_{\Omega} |\nabla v|^{2}dx+\alpha \left|\ds\int_{\Omega}|v|v\,dx \right|}{\ds\int_{\Omega} |v|^{2}dx}$ ¶ 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.

11.
On the first Dirichlet Laplacian eigenvalue of regular polygons - Nitsch, Carlo
The Faber-Krahn inequality in R^{2} 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...

12.
A note on Petty's theorem - Marini, Michele; de Philippis, Guido
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.

13.
On the profile of solutions with time-dependent singularities for the heat equation - Kan, Toru; Takahashi, Jin
Let N ≥ 2, T $\in$ (0,∞] and ξ $\in$ C(0,T; R^{N}). Under some regularity condition for ξ, it is known that the heat equation ¶ u_{t} − Δu = 0, x $\in$ R^{N} \ {ξ(t)}, t $\in$ (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.

14.
Prices in the utility function and demand monotonicity - Barucci, Emilio; Gazzola, Filippo
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.

15.
Lower bounds for blow-up time in a parabolic problem with a gradient term under various boundary conditions - Marras, Monica; Vernier Piro, Stella; Viglialoro, Giuseppe
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.

16.
Conservation of the mass for solutions to a class of singular parabolic equations - Fino, Ahmad Z.; Düzgün, Fatma Gamze; Vespri, Vincenzo
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}

18.
Nonexistence and existence results for a 2nth-order discrete Dirichlet boundary value problem - Shi, Haiping; Liu, Xia; Zhang, Yuanbiao
This paper is concerned with a 2nth-order nonlinear difference equation. By making use of the critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for Dirichlet boundary value problem and give some new results. The existing results are generalized and significantly complemented.

19.
Attractors of iterated function systems and associated graphs - Dumitru, Dan; Mihail, Alexandru
The aim of this article is to establish some conditions under which the attractors of iterated function systems become dendrites. We associate to an attractor of an iterated function system (IFS) some graphs and we prove that for a large class of IFSs their attractors are dendrites if the associated graphs are trees. We also give some examples of such sets.

20.
Corrections to "Extremal disks and extremal surfaces of genus three" - Nakamura, Gou
We correct a result in "Extremal disks and extremal surfaces of genus three", Kodai Math. J. 28, no. 1 (2005), 111-130. In the paper we have shown that there exist 16 compact Riemann surfaces of genus three up to conformal equivalence in which two extremal disks are isometrically embedded. However we have three more of them up to conformal equivalence. In the present paper we give these three surfaces and show that they are hyperelliptic. We also determine the groups of automorphisms of them.