We describe recent results on the regularity for the Stefan problem obtained in a joint work with... more We describe recent results on the regularity for the Stefan problem obtained in a joint work with I. Athanasopoulos and L. Caffarelli.
We consider the problem where G⊂R + is a bounded domain, adjacent to the x-axis, k is a real posi... more We consider the problem where G⊂R + is a bounded domain, adjacent to the x-axis, k is a real positive number, A is a positive definite 2x2 matrix with L entries. Using symmetrization techniques, we get the sharp form of some estimates for u. The main tool is an isoperimetric inequality in the upper half plane with respect to the
In this paper we give an overview of some recent and older results concerning free boundary probl... more In this paper we give an overview of some recent and older results concerning free boundary problems governed by elliptic operators.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Proceedings of the American Mathematical Society, Feb 1, 1982
We exhibit an example which proves that the elliptic measure for a second-order operator of the f... more We exhibit an example which proves that the elliptic measure for a second-order operator of the form 2";-X OaD» with continuous coefficients can be singular with respect to the surface measure on the boundary of a smooth two-dimensional domain.
The obstacle problem for a fractional power of the Laplace operator appears in many contexts, suc... more The obstacle problem for a fractional power of the Laplace operator appears in many contexts, such as in the study of anomalous diffusion [5], in the so called quasi-geostrophic flow problem [12], and in pricing of American options governed by assets evolving according to jump processes [13].
One proves the existence of a body G, whose density is known, which creates on its surface a give... more One proves the existence of a body G, whose density is known, which creates on its surface a given potential.
We present some recent results concerning free or moving boundary problems for parabolic operator... more We present some recent results concerning free or moving boundary problems for parabolic operators. We also single out some questions that are still open, also adding some clues about the typical difficulties one has to face by trying to get an answer.
Annali di Matematica Pura ed Applicata, Dec 1, 1981
Sunto.-Iq~ questo lavoro vengono lorese in eonsiderazione soluzioq~i deboli q~on-negative di eqq~... more Sunto.-Iq~ questo lavoro vengono lorese in eonsiderazione soluzioq~i deboli q~on-negative di eqq~azioni loaraboliehe liq~eari del seeondo ordine in/orma di divergenza in un dominie eili/adrieo di Rn• R. Per tali soluzioni si stabilisee q~n principio di ripe Harnaek che si estende sine alla #ontiera del dominie. Si l~rova poi ehe tale principle ~ equivalente ad una stima uni-]orme rer la misura ealor@a assoeiata ad ogni ol~eratbre del ripe suddetto. Per soluzioni continue q~elta ehiusura del dominie ~ poi l~rOvata wan versione 1~i~ /orte del l~rincil~io di eui sopra.
We use Perron method to construct a weak solution to a two-phase free boundary problem with right... more We use Perron method to construct a weak solution to a two-phase free boundary problem with right-hand-side. We thus extend the results in [C3] for the homogeneous case.
2.1. The problem is analyzed in Section 2.3.3. Its solution is given by (2.34): $$ \rho \left(x,t... more 2.1. The problem is analyzed in Section 2.3.3. Its solution is given by (2.34): $$ \rho \left(x,t\right)=\Big\{\begin{array}{l}\begin{array}{cc} {\rho}_m & \kern2.52em for\kern0.6em x\le -{v}_mt \end{array}\\ \begin{array}{cc} \frac{\rho_m}{2}\left(1-\frac{x}{v_mt}\right) & for-{v}_mt<x<{v}_mt, \end{array}\\ \begin{array}{cc} 0 & \kern3.12em for\kern0.5em x\ge {v}_mt \end{array}\end{array} $$
In this chapter we shall focus on models in which reaction and diffusion are in competition. Of p... more In this chapter we shall focus on models in which reaction and diffusion are in competition. Of particular interest is the study of the asymptotic behavior of the solutions as time goes on and to explore the existence and the stability properties of limiting steady states.
We describe recent results on the regularity for the Stefan problem obtained in a joint work with... more We describe recent results on the regularity for the Stefan problem obtained in a joint work with I. Athanasopoulos and L. Caffarelli.
We consider the problem where G⊂R + is a bounded domain, adjacent to the x-axis, k is a real posi... more We consider the problem where G⊂R + is a bounded domain, adjacent to the x-axis, k is a real positive number, A is a positive definite 2x2 matrix with L entries. Using symmetrization techniques, we get the sharp form of some estimates for u. The main tool is an isoperimetric inequality in the upper half plane with respect to the
In this paper we give an overview of some recent and older results concerning free boundary probl... more In this paper we give an overview of some recent and older results concerning free boundary problems governed by elliptic operators.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Proceedings of the American Mathematical Society, Feb 1, 1982
We exhibit an example which proves that the elliptic measure for a second-order operator of the f... more We exhibit an example which proves that the elliptic measure for a second-order operator of the form 2";-X OaD» with continuous coefficients can be singular with respect to the surface measure on the boundary of a smooth two-dimensional domain.
The obstacle problem for a fractional power of the Laplace operator appears in many contexts, suc... more The obstacle problem for a fractional power of the Laplace operator appears in many contexts, such as in the study of anomalous diffusion [5], in the so called quasi-geostrophic flow problem [12], and in pricing of American options governed by assets evolving according to jump processes [13].
One proves the existence of a body G, whose density is known, which creates on its surface a give... more One proves the existence of a body G, whose density is known, which creates on its surface a given potential.
We present some recent results concerning free or moving boundary problems for parabolic operator... more We present some recent results concerning free or moving boundary problems for parabolic operators. We also single out some questions that are still open, also adding some clues about the typical difficulties one has to face by trying to get an answer.
Annali di Matematica Pura ed Applicata, Dec 1, 1981
Sunto.-Iq~ questo lavoro vengono lorese in eonsiderazione soluzioq~i deboli q~on-negative di eqq~... more Sunto.-Iq~ questo lavoro vengono lorese in eonsiderazione soluzioq~i deboli q~on-negative di eqq~azioni loaraboliehe liq~eari del seeondo ordine in/orma di divergenza in un dominie eili/adrieo di Rn• R. Per tali soluzioni si stabilisee q~n principio di ripe Harnaek che si estende sine alla #ontiera del dominie. Si l~rova poi ehe tale principle ~ equivalente ad una stima uni-]orme rer la misura ealor@a assoeiata ad ogni ol~eratbre del ripe suddetto. Per soluzioni continue q~elta ehiusura del dominie ~ poi l~rOvata wan versione 1~i~ /orte del l~rincil~io di eui sopra.
We use Perron method to construct a weak solution to a two-phase free boundary problem with right... more We use Perron method to construct a weak solution to a two-phase free boundary problem with right-hand-side. We thus extend the results in [C3] for the homogeneous case.
2.1. The problem is analyzed in Section 2.3.3. Its solution is given by (2.34): $$ \rho \left(x,t... more 2.1. The problem is analyzed in Section 2.3.3. Its solution is given by (2.34): $$ \rho \left(x,t\right)=\Big\{\begin{array}{l}\begin{array}{cc} {\rho}_m & \kern2.52em for\kern0.6em x\le -{v}_mt \end{array}\\ \begin{array}{cc} \frac{\rho_m}{2}\left(1-\frac{x}{v_mt}\right) & for-{v}_mt<x<{v}_mt, \end{array}\\ \begin{array}{cc} 0 & \kern3.12em for\kern0.5em x\ge {v}_mt \end{array}\end{array} $$
In this chapter we shall focus on models in which reaction and diffusion are in competition. Of p... more In this chapter we shall focus on models in which reaction and diffusion are in competition. Of particular interest is the study of the asymptotic behavior of the solutions as time goes on and to explore the existence and the stability properties of limiting steady states.
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