Analysis and Partial Differential Equations 2024

 

OIST-Oxford-SLMath Summer Graduate School on Analysis of Partial Differential Equations to be held in 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. 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-theoretic 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 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 of 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 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 criteria 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: 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 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;

​Monday, July 29

• 9:00-10:15. Course II – Lecture 1. Ugur Abdulla, Harmonic functions - the mathematical representation of physical quantities in equilibrium
• 10:30-11:45. Course II – Lecture 2/active learning session (als). Ugur Abdulla, Fundamental harmonic function, representation formula for the gravitational potential via Poisson PDE
• 12:00-13:00. Lunch
• 13:00-14:15. Course II - Lecture 3. Ugur Abdulla, Mean value formula and strong maximum principle.
• 14:15-14:45. Tea and Coffee
• 14:45-16:00. Course II – Lecture 4/als. Ugur Abdulla, Green's function and Poisson integral formula

 

Tuesday, July 30

 

• 9:00-10:15. Course I – Lecture 1. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form I
• 10:30-11:45. Course I – Lecture 2/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form II
• 12:00-13:00. Lunch
• 13:00-14:15. Course I - Lecture 3. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form III
• 14:15-14:45. Tea and Coffee
• 14:45-16:00. Course I – Lecture 4/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form IV

Wednesday, Jujy 31

　

• 9:00-10:15. Course II – Lecture 5. Ugur Abdulla, Harnack inequality, interior estimates of derivatives and convergence theorems.
• 10:30-11:45. Course II – Lecture 6/ als. Ugur Abdulla, Superharmonic functions

• 12:00-13:00. Lunch
• 13:00-14:00 Plenary Lecture: Gui-Qiang Chen, TBA
• 14:00-14:30 Tea and Coffee
• 14:30-15:45. Course I - Lecture 5. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form V
• 16:00-17:00. Course I – Lecture 6/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form VI

Thursday, August 1

• 9:00-10:15. Course II – Lecture 7. Ugur Abdulla, Superharmonic function minimum principle. Perron's method
• 10:30-11:45. Course II – Lecture 8/ als. Ugur Abdulla, Boundary regularity of harmonic functions. The criterion for uniqueness.
• 12:00-13:00. Lunch
• 13:00-14:15. Course I - Lecture 7. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form VII
• 14:15-14:45 Tea and Coffee
• 14:45-16:00. Course I – Lecture 8/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form VIII

Friday, August 2

• 9:00-10:15. Course II – Lecture 9. Ugur Abdulla, Boundary regularity and barrier, Bouligand's theorem.
• 10:30-11:45. Course II – Lecture 10/ als. Ugur Abdulla, Capacity and Newtonian potentials
• 12:00-13:00. Lunch
• 13:00-14:00 Plenary Lecture: Juan Manfredi, TBA
• 14:00-14:30 Tea and Coffee
• 14:30-15:45. Course I - Lecture 9. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form IX
• 16:00-17:00. Course I – Lecture 10/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form X

Saturday, August 3

• Field trip to Naha, Okinawa.

Sunday, August 4

• Free day

Monday, August 5

• 9:00-10:15. Course I - Lecture 11. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XI
• 10:30-11:45. Course I – Lecture 12/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XII
• 12:00-13:00. Lunch
• 13:00-14:00 Plenary Lecture: Lawrence C Evans, Weak convergence methods for the nonlinear PDEs I
• 14:00-14:30 Tea and Coffee
• 14:30-15:45. Course II – Lecture 11. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XI
• 16:00-17:00. Course II – Lecture 12/als. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XII

Tuesday, August 6

• 9:00-10:15. Course I - Lecture 13. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XIII
• 10:30-11:45. Course I – Lecture 14/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XIV
• 12:00-13:00. Lunch
• 13:00-14:00 Plenary Lecture: Lawrence C Evans, Weak convergence methods for the nonlinear PDEs II
• 14:00-14:30 Tea and Coffee
• 14:30-15:45. Course II – Lecture 13. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XIII
• 16:00-17:00. Course II – Lecture 14/als. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XIV

Wednesday, August 7

• 9:00-10:15. Course II - Lecture 15. Ugur Abdulla, Peron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XV
• 10:30-11:45. Course I – Lecture 15. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XV
• 12:00-13:00. Lunch
• 13:00-14:00 Plenary Lecture: Ugur G. Abdulla, Kolmogorov Problem and Wiener-type criteria for the Removability of Fundamental Singularity for the Parabolic PDEs
• 14:30-21:00 Field trip to Okinawa Aquarium

Thursday, August 8

• 9:00-10:15. Course I - Lecture 16. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XVI
• 10:30-11:45 Course I – Lecture 17/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XVII
• 12:00-13:00. Lunch
• 13:00-14:15 . Course II – Lecture 16. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XVI
• 14:15-14:45 Tea and Coffee
• 14:45-16:00. . Course II – Lecture 17/als. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XVII

Friday, August 9

• 9:00-10:15. Course I - Lecture 18. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XVIII
• 10:30-11:45 Course I – Lecture 19/als. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XIX
• 12:00-13:00. Lunch
• 13:00-14:15 . Course II – Lecture 18. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XVIII
• 14:15-14:45 Tea and Coffee
• 14:45-16:00. . Course II – Lecture 19/als. Ugur Abdulla, Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XIX

 

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

 

1. Measure Theory and BV Functions.

          Radon and Hausdorff measures, week convergence of measures,

convolutions and representation of BV functions.

 

2. Sets of Finite Perimter.

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.

 

3. 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 conservations laws.

 

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

 

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, Bouligand’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:

1. Basic Measue Theory,  Distribution Theory, Sobolve Spaces, Functional Analysis
2. 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

 

Title and Abstract of Plenary Lectures

July 31 (Wed) 13:00 - 14:00 . Prof. Gui-Qiang G. Chen (University of Oxford)

Title:  Entropy Analysis and Singularities of Solutions for Nonlinear Hyperbolic Conservation Laws

Abstract:  In this talk, we present some reflections and recent developments in solving several longstanding open problems involving the singularities of entropy solutions for nonlinear hyperbolic conservation laws and related nonlinear partial differential equations through entropy analysis and associated methods. These problems especially include the minimal entropy conditions for well-posedness, the understanding of the phenomena of cavitation/decavitation and concentration/deconcentration, and the rigorous analysis for entropy solutions via the theory of divergence-measure fields, among others. Further related topics, perspectives, and open problems will also be addressed.

 

Aug.2(Fro) 13:00 - 14:00 Prof. Juan J. Manfredi (University of Pittsburgh)

Aug. 7 (Wed) 13:00 - 14:00 Prof. Ugur Abdulla （OIST)

Title: Kolmogorov Problem and Wiener-type Criteria for the Removability of the Fundamental Singularity for the Elliptic and Parabolic PDEs 

Abstract: Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results of 20th-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, probability and measure theories. This talk will address the major problem in the Analysis of PDEs on the nature of singularities reflecting the natural phenomena.  I will present my solution of the Kolmogorov's Problem (1928) expressed in terms of the new Wiener-type criterion for the removability of the fundamental singularity for the heat equation. The new concept of regularity or irregularity of singularity point for the parabolic (or elliptic) PDEs is defined according to whether or not the caloric (or harmonic) measure of the singularity point is null or positive. The new Wiener-type criterion precisely characterizes the uniqueness of boundary value problems with singular data, reveal the nature of the harmonic or caloric measure of the singularity point, asymptotic laws for the conditional Markov processes, and criteria for thinness in minimal-fine topology. The talk will end with the description of some outstanding open problems and perspectives of the development of the potential theory of nonlinear elliptic and parabolic PDEs.