Moving plane method
for p-Laplacian in Real
and Hyperbolic space

BGSMath Course

Moving plane method for p-Laplacian in Real and Hyperbolic space

BGSMath Course

From March 09, 2026
to March 26, 2026

Venue: IMUB - Universitat de Barcelona

Room: Seminar room

Registration deadline 05 / 03 / 2026

SCHEDULE

Introduction

Symmetry properties of solutions of partial differential equations is a very interesting and challenging subject. Most often if the solution admits certain symmetry, then it simplifies the PDE to a lower dimensional space and this often helps in having a better understanding of the solution. For example radial symmetry makes the PDE into an ODE and analysing this ODE is much simpler in many situations than studying the original PDE. These are the kind of tools used in many situations to establish classification results for solutions of PDEs or for proving uniqueness results.

There are many tools to establish the symmetry of positive solutions of PDEs which are of elliptic type. The most prominent one is the method of moving planes. The method was developed by Alexandrov to establish that the closed connected hypersurfaces of constant mean curvature in the Euclidean space are spheres. The method was introduced in the case of PDE by Serrin to study an over-determined problem and later developed further to study the symmetry properties of elliptic type PDEs by Gidas, Ni and Nirenberg. Since then the method has been used in a variety of contexts to study symmetry properties. The method relies heavily on the maximum principle or more generally strong comparison principle.

One of the significant applications of this method is in establishing the radial symmetry (and hence the classification) of solutions of the semilinear elliptic PDE:

$\Delta u = u^{\frac{n+2}{n-2}} \quad \text{in } \mathbb{R}^n, \qquad u > 0 \text{ in } \mathbb{R}^n, \int_{\mathbb{R}^n} |\nabla u|^2 \, dx < \infty.$

In fact this is the Euler–Lagrange equation corresponding to the Sobolev inequality in the space $D^{1,2}(\mathbb{R}^n)$ .

However when it comes to the Euler–Lagrange equation of the Sobolev inequality in the general space $D^{1,p}(\mathbb{R}^n)$ , it is the p-Laplace equation:

-\Delta_p u = u^{p^*-1} \quad \text{in } \mathbb{R}^n, \qquad u \ge 0 \text{ in } \mathbb{R}^n, \int_{\mathbb{R}^n} |\nabla u|^p \, dx < \infty.

where $p^* = \frac{np}{n-p}$ and $\Delta_p u = \operatorname{div}\bigl( |\nabla u|^{p-2} \nabla u \bigr).$ , which is quasilinear and degenerate elliptic and hence strong comparison principle is missing in this case to directly apply the moving plane method. Another issue in applying the moving plane method in the whole space $\mathbb{R}^n$ is to start the plane moving from infinity.

In the past decade, tools have been developed to tackle these issues and prove the symmetry results for p-Laplace equations in the Euclidean space $\mathbb{R}^n$ and in the hyperbolic space $\mathbb{H}^n$ . In this series of lectures we plan to discuss these issues.

lecturers

Kunnath Sandeep

TIFR Centre for Applicable Mathematics

Professor and Dean at the TIFR Centre for Applicable Mathematics, Bangalore (India). His research focuses on nonlinear partial differential equations, variational methods, and problems involving critical exponents in Euclidean and hyperbolic spaces. He is known for fundamental contributions to symmetry results in elliptic PDEs and to sharp inequalities of Sobolev and Moser–Trudinger type. Prof. Sandeep received the Shanti Swarup Bhatnagar Prize (2015), one of India’s highest science awards, and he was elected in 2019 as Fellow of the Indian Academy of Sciences.

Course Content

The following topics will be covered :
1- Classical Moving plane method and the case p = 2 ( [1], [2])
2- Regularity estimates and comparison principle for p ∈(1, n) ([3], [5], [6] )
3- Symmetry of Solutions of the Euler Lagrange equation corresponding to the Sobolev inequality ([8],[7])
4- Moving plane method for p-Laplace type equations in Hyperbolic space. ([4], [9])
Prerequisites : Knowledge of basic theory of second order uniformly elliptic PDEs, like maximum principle, comparison principle etc.

SCHEDULE

Schedule: Six sessions, each two hours long

Monday, March 9, 11:00–13:00
Thursday, March 12, 11:00–13:00
Monday, March 16, 11:00–13:00
Thursday, March 19, 11:00–13:00
Monday, March 23, 11:00–13:00
Thursday, March 26, 11:00–13:00

BIBLIOGRAPHY

[1] Gidas, B. ; Ni, Wei Ming and Nirenberg, L. Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209-243.
[2] Serrin, James A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304-318.
[3] Damascelli, Lucio. Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 4, 493-516.
[4] Almeida, Luís; Damascelli, Lucio; Ge, Yuxin. A few symmetry results for nonlinear elliptic PDE on noncompact manifolds Ann. Inst. H. Poincaré C Anal. Non Linéaire 19 (2002), no. 3, 313-342.
[5] Damascelli, Lucio; Sciunzi, Berardino. Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations J. Differential Equations 206 (2004), no. 2, 483-515.
[6] Damascelli, Lucio; Sciunzi, Berardino. Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations 25 (2006), no. 2, 139-159.
[7] Vétois, Jérôme A priori estimates and application to the symmetry of solutions for critical p -Laplace equations J. Differential Equations 260 (2016), no. 1, 149-161.
[8] Damascelli, Lucio; Merchán, Susana; Montoro, Luigi; Sciunzi, Berardino. Radial symmetry and applications for a problem involving the −∆p operator and critical nonlinearity in \mathbb{R}^N. Adv. Math. 265 (2014)313-335.
[9] Ramya Dutta, Kunnath Sandeep. Symmetry for a quasilinear elliptic equation in hyperbolic space. Ann. Scuola Norm. Sup. Pisa Cl. Sci. To appear.

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