数学研究所

学术报告

偏微分方程研讨班

Speaker: Prof. Nicola Fusco(University of Naples Federico II, Italy)

Inviter: 张翼 副研究员

Language: English

Title:Isoperimetric type inequalities for capillary surfaces （1）

Time&Venue: 2025年6月2日（周一）9:00–9:50&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

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Title:Isoperimetric type inequalities for capillary surfaces （2）

Time&Venue: 2025年6月2日（周一）10:00–11:00&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

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Title:Isoperimetric type inequalities for capillary surfaces （3）

Time&Venue: 2025年6月3日（周二）9:00–9:50&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

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Title:Isoperimetric type inequalities for capillary surfaces （4）

Time&Venue: 2025年6月3日（周二）10:00–11:00&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

Title:Isoperimetric type inequalities for capillary surfaces （5）

Time&Venue: 2025年6月5日（周四）9:00–9:50

&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

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Title:Isoperimetric type inequalities for capillary surfaces （6）

Time&Venue: 2025年6月5日（周四）10:00–11:00

&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

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Title:Isoperimetric type inequalities for capillary surfaces （7）

Time&Venue: 2025年6月6日（周五）9:00–9:50

&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

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Title:Isoperimetric type inequalities for capillary surfaces （8）

Time&Venue: 2025年6月6日（周五）10:00–11:00

&南楼N913

Abstract:  The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.