The 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 )

        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 ∞ 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 ∞ defined according as to the harmonic (or parabolic) measure of ∞ is null or positive, and discussion of the proof of the new Wiener criterion for the regularity of ∞. The Wiener test at ∞ 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 ∞ arises as a thinness criterion at ∞ in fine topology. In a probabilistic context, the Wiener test at ∞ 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.

For more information, please visit the website.
Workshop website: https://www.oist.jp/conference/2026-summer-graduate-school-analysis-and-pde

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