Numerical Method for Solving an Inverse Problem for Laplace’s Equation in a Domain with an Unknown Inner Boundary

Computational Mathematics and Mathematical Physics - An inverse problem for Laplace’s equation in a doubly connected two-dimensional domain is considered. Given Dirichlet and Neumann data...     

Proof of Stability in the Brower–Paul Problem

Doklady Mathematics - We study the stability of equilibrium in the problem known as “a ball on a rotating saddle,” which was first considered by the famous Dutch mathematician Brauer in...     

Leibniz’s Ontological Proof of the Existence of God and the Problem of »Impossible Objects«

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premise

1. (1)
   God is possible.
    

Second, a demonstration of the “remarkable proposition”

1. (2)
   If God is possible, then God exists.
    

The present contribution investigates an interesting paper in which Leibniz tries to prove proposition (2). It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ (denoting the concept of existence) as follows:

1. (3)
   \(g=_{\mathrm{df}} \,\upiota x\square E(x)\).
    

Proposition (2) which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows:

1. (4)
   \(\diamondsuit E(\upiota x\square E(x)) \rightarrow \, E(\upiota x\square E(x))\).
    

At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to (4), the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \(\hbox {F}(y)\) and \(\lnot F(y)\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of (2) turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of (1), i.e. the self-consistency of the concept of a necessary being, remains to be proven.

     

Krein’s Theorems for a Dissipative Boundary Value Transmission Problem

Let us consider the differential expression $$\ell (y)=-y^{\prime \prime }+q(x)y,\quad x\in I:=[0,c)\cup (c,\infty ),$$ where c is a transmission point and is regular for the differential expression ?(y). We assume that Weyl’s limit-circle case holds for the differential expression ?(y) on I. In this paper, using Krein’s theorems, we investigate the completeness of the root vectors of a singular dissipative boundary value transmission problem generated by ?(y).     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

A Free Boundary Problem for an Elliptic–Parabolic System: Application to a Model of Tumor Growth

Abstract

In this article, we study a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic. The system models the growth of a tumor with arbitrary initial shape. We establish the existence and uniqueness of a solution for some time interval. In the special case where we only have the elliptic equation, the problem coincides with the Hele–Shaw problem.     

Fujita Exponent for Porous Medium Equation with Convection and Nonlinear Boundary Condition

This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that determines the critical exponent.     

Global Existence for a Parabolic-hyperbolic Free Boundary Problem Modelling Tumor Growth

In this paper we study a free boundary problem modelling tumor growth, proposed by A. Friedman in 2004. This free boundary problem involves a nonlinear second-order parabolic equation describing the diffusion of nutrient in the tumor, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells, respectively. By applying Lp theory of parabolic equations, the characteristic theory of hyperbolic equations, and the Banach fixed point theorem, we prove that this problem has a unique global classical solution.     

A Free Boundary Problem for ODE Arising from Degenerate Parabolic Equations

Assume B∈C[0,+∞)∩)C~1(0,+∞) and B(υ) is strictly increasingand concave. That the free boundary Problem for ODE υ″=-1/2s〔B(υ)〕′for s>s_0, υ(s_0), υ′(s_0)=-(a/2)s_0, limυ(s)=βhas a unique solution (υ(s),s_0) with s_0∈〔-(2β/a)~(1/2),0) is proved. This problemarises, for example, from the investigation of the structure of discontinuous solutions for degenerate parabolic equations like(u/t)=A(u)/x~2     

Schwarz’s Lemma and Estimates of Coefficients in the Case of an Arbitrary Set of Boundary Fixed Points

Mathematical Notes -     

The Proof of a Theorem in DC Problem

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Immersed Solutions of Plateau’s Problem for Piecewise Smooth Boundary Curves with Small Total Curvature

We provide a new proof of the classical result that any closed rectifiable Jordan curve ${\Gamma \subset \mathbb{R}^3}$ being piecewise of class C ² bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Γ is smaller than 6π. In contrast to the methods due to Osserman (Ann Math 91(2):550–569, 1970), Gulliver (Ann Math 97(2):275–305, 1973) and Alt (Math Z 127:333–362, 1972, Math Ann 201:33–35, 1973), our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau’s problem for polygonal boundary curves, provided by the first author’s accomplishment (The Plateau problem, Fuchsian equations and the Riemann–Hilbert problem. Mémoires de la Soc. Math. Fr. (to appear) arXiv: 1003.0978) of Garnier’s ideas in (Annales scientifiques de l’É.N.S. 45:53–144, 1928).     

Estimates in Hölder Classes for the Solution of an Inhomogeneous Dirichlet Problem for a Singularly Perturbed Homogeneous Convection–Diffusion Equation

Computational Mathematics and Mathematical Physics - An inhomogeneous Dirichlet boundary value problem for a singularly perturbed homogeneous convection–diffusion equation with constant...     

On the Exceptional Set of Goldbach’s Problem in Short Intervals

Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant such that for we have .     

Stability in an overdetermined problem for the Green’s function

In the plane, we consider the problem of reconstructing a domain from the normal derivative of its Green’s function (with fixed pole) relative to the Dirichlet problem for the Laplace operator. By means of the theory of conformal mappings, we derive stability estimates of Hölder type.     

The Problem of Finding the Kernels in the System of Integro-Differential Maxwell’s Equations

Journal of Applied and Industrial Mathematics - We pose the direct and inverse problems of finding the electromagnetic field and the diagonal memory matrix for the reduced canonical system of...     

Chebyshev’s Problem of the Moments of Nonnegative Polynomials

Mathematical Notes - We study the problem of P. L. Chebyshev (proposed in 1883) concerning the extreme values of moments of nonnegative polynomials with weight on the interval $$[-1,1]$$ at a fixed...     

On the Existence of Solutions for a Class of Singular Boundary Value Problem

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