TY - CHAP

T1 - Stable solutions to some elliptic problems

T2 - Minimal cones, the Allen-Cahn equation, and blow-up solutions

AU - Cabré, Xavier

AU - Poggesi, Giorgio

PY - 2018/10/4

Y1 - 2018/10/4

N2 - These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19–23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar. The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J. Simons on the flatness of stable minimal cones in low dimensions. Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E. De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone. The third problem was raised by H. Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic. Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces.

AB - These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19–23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar. The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J. Simons on the flatness of stable minimal cones in low dimensions. Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E. De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone. The third problem was raised by H. Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic. Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces.

UR - http://www.scopus.com/inward/record.url?scp=85054486007&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-95186-7_1

DO - 10.1007/978-3-319-95186-7_1

M3 - Chapter

AN - SCOPUS:85054486007

SN - 9783319951850

T3 - Lecture Notes in Mathematics

SP - 1

EP - 45

BT - Geometry of PDEs and Related Problems

A2 - Bianchini, Chiara

A2 - Henrot, Antoine

A2 - Magnanini, Rolando

PB - Springer-Verlag Wien

CY - Austria

ER -