# OIST-Oxford-SLMath Summer School 2024

**OIST-Oxford-SLMath Summer Graduate School on Analysis of Partial Differential Equations to be held on July 29 – August 9, 2024, at Okinawa, Japan**

**Organizing committee: **

- Prof. Ugur G. Abdulla, Professor and Head of the Analysis & PDE Unit, Okinawa Institute of Science and Technology (OIST), Okinawa, Japan
- Prof. Gui-Qiang G. Chen, Statutory Professor in the Analysis of PDEs, Director of the Oxford Centre for Nonlinear PDEs (OxPDE), University of Oxford, United Kingdom

**Lecturers: **

- Prof. Ugur G. Abdulla, Professor and Head of the Analysis & PDE Unit, Okinawa Institute of Science and Technology (OIST), Okinawa, Japan
- Prof. Gui-Qiang G. Chen, Statutory Professor in the Analysis of PDEs, Director of the Oxford Centre for Nonlinear PDEs (OxPDE), University of Oxford, United Kingdom
- Prof. Alessia E Kogoj, Associate Professor of Mathematical Analysis, University of Urbino, Italy
- Prof. Monica Torres, Professor of Mathematics, Purdue University, USA

**Plenary Lecturers:**

- Prof. Ugur G. Abdulla, Professor and Head of the Analysis & PDE Unit, Okinawa Institute of Science and Technology (OIST), Okinawa, Japan
- Prof. Gui-Qiang G. Chen, Statutory Professor in the Analysis of PDEs, Director of the Oxford Centre for Nonlinear PDEs (OxPDE), University of Oxford, Oxford, United Kingdom
- Prof. L. Craig Evans, University of California, Berkeley, USA
- Prof. Juan Manfredi, University of Pittsburgh, Pittsburgh, USA

**Teaching Assistants: **

- Dr. Federica Gregorio, University of Salerno, Italy
- Dr. Daniel Tietz, Postdoctoral Scientist, Analysis & PDE Unit, OIST

**Scientific Description: **SGS will offer two mini-courses during the two-week long school:

- Course I:
**Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form**(Gui-Qiang G. Chen & Monica Torres)

In this course, we will present some recent developments in the theory of divergence-measure fields via measure-theoretical analysis and its applications to the analysis of nonlinear PDEs of conservative form – nonlinear conservation laws. We plan to start with an introduction to measure theory, BV functions, and a set of finite perimeter, and then present the theory of divergence-measure fields in L^{∞}, L^{p}, and the space of Radon measures, respectively. With these, we will discuss applications of the theory of divergence-measure fields in several fundamental research directions, including the mathematical formulation of the balance law and derivation of systems of balance laws via the Cauchy fluxes, and the analysis of entropy solutions of nonlinear conservation laws (especially, nonlinear hyperbolic conservation laws), among others. Some further developments, open problems, and current trends on the research topics will also be addressed.

**References**

- Chen, G.-Q. and Torres, M. Lecture Notes (
*to be available for the summer school*). - Chen, G.-Q. and Torres, M. (2021): Divergence-Measure Fields: Gauss-Green Formulas and Normal Traces.
*Notices Amer. Math. Soc.*68 (2021), no. 8, 1282–1290. - Chen, G.-Q. and Frid, H. (1999): Divergence-measure fields and hyperbolic conservation laws.
*Arch. Ration. Mech. Anal.*147(2): 89 – 118. - Chen, G.-Q., Torres, M., and Ziemer, W. P. (2009): Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws.
*Comm. Pure Appl. Math.*62(2): 242 – 304. - Chen, G.-Q., Comi, G. E., Torres, M. (2019): Cauchy fluxes and Gauss-Green formulas for divergence-measure fields over general open sets.
*Arch. Ration. Mech. Anal.*233: 87–166. - Dafermos, C. M. (2016): Hyperbolic Conservation Laws in Continuum Physics, 4th Ed.,
*Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],*325, Springer-Verlag: Berlin, 2016. - Evans, L. C. and Gariepy, R. F. (1992): Measure Theory and Fine Properties of Functions.
*Studies in Advanced Mathematics*. CRC Press: Boca Raton, FL, 1992. - Federer, H.: Geometric Measure Theory. Springer-Verlag New York Inc.: New York, 1969.

**Additional References:**

- L. Ambrosio, N. Fusco, and D. Pallara (2000): Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press: New York.
- Anzellotti, G. (1984): Pairings between measures and bounded functions and compensated compactness.
*Ann. Mat. Pura Appl.*(4), 135: 293–318. - Chen, G.-Q. and Frid, H. (2003): Extended divergence-measure fields and the Euler equations for gas dynamics.
*Commun. Math. Phys.*236: 251–280. - Chen, G.-Q. and M. Torres, M. (2005): Divergence-measure fields, sets of finite perimeter, and conservation laws.
*Arch. Ration. Mech. Anal*. 175: 245–267. - Maz'ya, V. G. (2011): Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer-Verlag: Berlin-Heidelberg.
- Pfeffer, W. F. (2012): The Divergence Theorem and Sets of Finite Perimeter, Chapman & Hall/CRC: Boca Raton, FL.

- Course II:
**Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs**(Ugur Abdulla & Alessia Kogoj)

Norbert Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results in XX-century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories. In this course, we will present some recent developments precisely characterizing the regularity of the point at \(\infty\) for second-order elliptic and parabolic PDEs and broadly extending the role of the Wiener test in classical analysis. We preface the description of the course with a citation from a classical paper by Wiener: *the Dirichlet Problem (DP) divides itself into two parts, the first of which is the determination of the harmonic function corresponding to certain boundary conditions, while the second is the investigation of the behavior of this function in the neighborhood of the boundary*. In the first week of the course, we focus on proving the existence of the solution to the DP for the Laplace equation, and its parabolic counterpart for the heat/diffusion equation. Solvability, in some generalized sense, of the Dirichlet problem in an arbitrary open set with prescribed data on its topological boundary is realized within the class of* resolutive boundary functions*, identified by Perron's method, and its Wiener and Brelot refinements. Such a method is referred to as the PWB method and the corresponding solutions are the PWB solutions. In the second week of the course, our focus will be on the boundary regularity of PWB solutions. The regularity of a boundary point is a problem of local nature and it depends on the measure-geometric properties of the boundary in the neighborhood of the boundary point and the differential operator. After introducing the concept of Newtonian capacity we discuss Wiener’s celebrated criterion which expresses the boundary regularity in terms of the divergence of the Wiener integral with the integrand being a capacity of the exterior set in the neighborhood of the boundary point. The high point of the course is the new concept of regularity or irregularity of the point at \(\infty\) defined according as to the harmonic (or parabolic) measure of \(\infty\) is null or positive, and discussion of the proof of the new Wiener criterion for the regularity of \(\infty\). The Wiener test at \(\infty\) arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at \(\infty\) arises as a thinness criterion at \(\infty\) in fine topology. In a probabilistic context, the Wiener test at \(\infty\) characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criteria for the removability of singularities and/or for unique continuation at the finite boundary point.

**Participants: **The school will be well suited for an audience of graduate students with a wide range of abilities and knowledge. Both courses are designed in a way that the first week’s material mostly overlaps with the material taught in the standard graduate courses on Analysis and PDEs. Therefore, even first-year graduate students with solid undergraduate math backgrounds will have the potential for a comfortable and engaging start. Both courses will offer *a tour de force *for the transition from standard material to cutting-edge discoveries in the frontline of the field of PDEs. The structure of the courses will aim to create an active learning environment through a combination of classical and flipped classroom teaching. Every working day each course will present a classical-style lecture given by the professor followed by an active learning session. The goal of the second lecture is twofold: first to sharpen the comprehension of the material of the given lecture, and second, to prepare students for the forthcoming lecture. Both professors and assistants will be involved in active learning sessions to help students. Starting from day 2, active learning sessions will include student presentations of the assignments given in the previous day’s lecture. To address the variance of academic backgrounds, students will be divided into groups, each group including students with varying backgrounds. Each group will have at least one presentation during the course, which will include the participation of all group members. In order to make active learning sessions more effective an online discussion forum will be created, and students will be encouraged to post their questions and comments following every lecture. This discussion forum will define the major discussion topics of the following active learning sessions.

**References **

- U.G. Abdulla, Wiener’s Criterion at ∞ for the Heat Equation, Advances in Differential Equations, 13(5-6), (2008), 457-488.
- U.G. Abdulla, Wiener’s Criterion for the Unique Solvability of the Dirichlet Problem in Arbitrary Open Sets with Non-Compact Boundaries, Nonlinear Analysis, 67(2), (2007), 563-578.
- U.G. Abdulla, Regularity of ∞ for Elliptic Equations with Measurable Coefficients and Its Consequences, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 32, 10(2012), 3379-3397.
- U.G. Abdulla, Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and its Consequences
*,*Calculus of Variations and Partial Differential Equations, 57, (6), (2018), 57-157. - U.G. Abdulla, First Boundary Value Problem for the Diffusion Equation. I. Iterated Logarithm Test for the Boundary Regularity and Solvability, SIAM J. Math. Anal., 34(6), (2003), 1422–1434.
- U.G. Abdulla, Well-posedness of the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains, Trans. Amer. Math. Soc., 357(1), (2005), 247–265.
- U.G. Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, J. Math. Anal. Appl., 246, 2, 2001, 384-403
- D.H. Armitage and S.J. Gardiner, Classical Potential Theory, Springer Monographs in Mathematics, Springer, 2001.
- H. Bauer, Harmonische Raume und ihre Potentialtheorie, Lecture Notes in mathematics, Springer, 1966.
- M.Brelot, Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, 1967.
- J.L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, 1984.
- L.C. Evans and R.F. Gariepy, Wiener’s criterion for the heat equation, Arch. Ration. Mech. Anal., 78, 1982, 293-314.
- L.C. Evans, Partial Differential Equations, AMS, 2nd edition, 2010.
- L.L. Helms, Potential Theory, Universitext, Springer, 2009.
- E. Lanconelli, Sul problema di Dirichlet per l’equazione del cslore, Ann. Math. Pura Appl., 97, 1973, 83-114.
- I.G. Petrowsky, Zur Ersten Randwertaufgabe der Warmeleitungsgleichung, Composito Math., 1, 1935, 383-419.
- N. A. Watson, Introduction to Heat Potential Theory, Mathematical Surveys and Monographs, vol. 182, Amer. Math. Soc., Providence RI, 2012.
- N. Wiener, Certain notions in potential theory, J. Math. Phys., 3, 1924, 24-51.
- N. Wiener, The Dirichlet problem, J. Math. Phys., 3, 1924, 127-146.

**Lesson plan/Syllabus: **Daily schedule of the SGS in week 1 will be as follows;

- 9:30-10:45. Course 1 – Lecture 1.
- 10:45-12:00. Problem Session led by lecturers and postdocs.
- 12:30-1:30. Lunch
- 1:45-3:00. Course 2 – Lecture 1.
- 3:00-3:30. Tea and Coffee.
- 3:30-5:00. Problem Session led by lecturers and postdocs.

Order of the courses 1 and 2 will be replaced in week 2.

**Course 1:**** Topics covered during the 1st** **week (Gui-Qiang G. Chen & Monica Torres):**

**Measure Theory and BV Functions.**

** **Radon and Hausdorff measures, week convergence of measures, convolutions and representation of BV functions.

**Sets of Finite Perimeter.**

Basic properties of sets of finite perimeter, structure of sets of finite perimeter, almost one-sided smooth approximation of sets of finite perimeter, the approximation of sets of finite perimeter with respect to any measure that is absolutely continuous with respect to the co-dimension-one Hausdorff measure, the main approximation results.

**Smooth One-Side Approximations to General Open Sets**.

Smooth one-side approximations of the boundaries of general open Sets, smooth regular one-side deformation of the boundaries of Lipschitz open sets.

4. **Divergence-Measure Fields.**

Basic properties, product rules for divergence-measure fields, etc.

**Course 1:**** Topics covered during the ****2nd week (Gui-Qiang G. Chen & Monica Torres):**

5**. Divergence-Measure Fields in L ^{∞}**

The Gauss-Green formula over sets of finite perimeter, the divergence-measures of jump sets via the normal traces, consistency of the normal traces with the classical traces, extensions of divergence-measure fields, Gauss-Green formula over general open sets.

**6. Divergence-Measure Fields in L ^{p} and the Space of Radon Measures**

Gauss-Green formula over Lipschitz domains, Gauss-Green formula over general open sets, etc.

**7.** **Cauchy Flux, Balance Laws, and Entropy Solutions**.

Cauchy fluxes and divergence-measure fields, mathematical formulation of the balance law and derivation of systems of balance laws, entropy solutions of hyperbolic conservation laws, applications of divergence-measure fields to conservation laws.

**Course 2:**** Topics covered during the**** 1st week ( Ugur Abdulla & Alessia Kogoj):**

**5. Perron’s method for the Laplacian****. **

** ***Harmonic functions, Gauss mean value formula, superharmonic functions, solution of the Dirichlet problem for Euclidean balls, the Poisson kernel, generalized solution of the Dirichlet problem in the sense of Perron-Wiener-Brelot, boundary behavior of the PWB solution, ouligand’s theorem.*

**6. Perron’s method for the Heat operator.**

** ***The parabolic maximum principle, mean value theorem for caloric, super- and subcaloric functions, the existence of a basis of resolutive open sets for the Heat operator, convergence theorems and parabolic Harnack inequality, PWB solutions for the parabolic Dirichlet problem. *

**Course 2:**** Topics covered during the ****2nd week (Ugur Abdulla):**

**7. Wiener criterion at **\(\infty\)** for the Laplace equation.**

** ***Newtonian potentials , Newtonian capacity, minimization of energy functional in Hilbert space setting, Wiener criterion at finite boundary points, geometric tests for boundary regularity, harmonic measure, regularity of *\(\infty\)

*, Wiener criterion for the regularity of*\(\infty\)

*, fine topology, asymptotic laws for the Brownian motion*

**8. Wiener criterion at **\(\infty\)**for the heat equation:**

** ***Thermal potentials, thermal capacity, geometric iterated logarithm test, parabolic measure, regularity of *\(\infty\)*, boundary Harnack inequality for the heat equation, proof of the Wiener criterion at *\(\infty\)* for the heat equation, measure-theoretical, topological and probabilistic consequences of the Wiener test at *\(\infty\)*.*

**Prerequisites: **

- Basic Measue Theory, Distribution Theory, Sobolve Spaces, Functional Analysis
- In the Graduate Textbook: Lawrence C. Evans, Partial Differential Equations, AMS, 2nd edition, 2010:

- Reviewing calculus facts outlined in Appendix C: Calculus
- Reviewing facts outlined in Appendices D and E: Fundational Analysis and Measure Theory
- Review Section 2.2. Laplace’s Equation; and Section 2.3. Heat Equation;
- Solve exercises 2-17 from Section 2.3 Problems.
- Review Section 2.4, Section 3, and Section 5

**Math Subject Clasification numbers**:

**Course 1:** Primary: 28C05; 26B20; 28A05; 26B12; 35L65; 35L67, 35L50, 76A02, 35D30; 76L05; Secondary: 28A75; 28A25; 26B05; 26B30; 26B40; 35M30; 35B35; 35B40; 74J40.

**Key Words:** divergence-measure fields; PDE of divergence form; nonlinear conservation laws; hyerbolic conservation laws; sets of finite perimeter; BV functions; approximation; sets with Lipschitz boundary;open sets; Cauchy flux; balance laws; entropy solutions; foundation of continuum mechanics.

**Course 2:** Primary: 35J05; 35J25; 35K05; 35K20; 31C05; 31C15; 31C40; 31D05; 31A15; Secondary: 60J45; 60J65;54C50; 30C85; 32U20.

**Key words:** potential theory; elliptic and parabolic PDEs; Laplace equation; heat equation; Dirichlet problem; super- and subharmonic functions; Wiener criterion; boundary regularity; regularity (or irregularity) of \(\infty\); caloric function; super- and subcaloric functions; harmonic measure; parabolic measure; capacity; Newtonian potential; thermal capacity; thermal potential; Radon measure; fine topology; Brownian motion; Wiener processes;

**Location description:** The description of the facilities for the SGS is given in the following link