The principal aim of the volume is gathering all the contributions given by the speakers (mini courses) and some of the participants (short talks) of the summer school "Modern Problems in PDEs and Applications" held at the Ghent Analysis and PDE Center from 23 August to 2 September 2023. The school was devoted to the study of new techniques and approaches for solving partial differential equations, which can either be considered or arise from the physical point of view or the mathematical perspective. Both sides are extremely important since theories and methods can be developed independently, aiming to gather each other in a common objective. The aim of the summer school was to progress and advance in the problems considered. Note that real-world problems and their applications are classical study trends in physical or mathematical modelling. The summer school was organised in a friendly atmosphere and synergy, and it was an excellent opportunity to promote and encourage the development of the subject in the community.

Includes minicourses taught by worldwide leading experts in analysis, PDEs and its applications Shares the research of some outstanding young researchers in various areas of analysis and PDEs Gathers high quality short research announcements, communications and summaries

Autorentext

Mariana Chatzakou is a postdoctoral researcher at Ghent University. She obtained her PhD in mathematics in 2020 at Imperial College London. Her research interests concern the areas of harmonic analysis, non-commutative analysis, pseudo-differential operators, functional inequalities on Lie groups, and PDEs with very strong singularities.Joel Restrepo is a postdoctoral researcher at the Ghent Analysis and PDE Center at Ghent University in Belgium. Previously, he was a postdoctoral researcher at different research centers eg. in Russia. He has carried out mathematical research in a variety of topics connecting different areas of mathematics. His research links several techniques in complex analysis, operator theory, harmonic analysis, PDEs, etc. He has participated in many international conferences, seminars and other related events and has been involved in the organization of several of these scientific projects as well. He was awarded by the International Society for Analysis, its Applications and Computation (ISSAC) with the life membership in 2019 in view of his achievements and contributions to the theory of weighted classes of delta-subharmonic functions and potential theory.

Michael Ruzhansky is a Senior Full Professor of Mathematics at Ghent University in Belgium, and a Professor of Mathematics at Queen Mary University of London in the United Kingdom. His research interests mainly lie in Partial Differential Equations, Microlocal and Harmonic Analysis, and Pseudo-Differential Operators on Lie Groups and Manifolds. Previously, he had appointments at Utrecht University, Johns Hopkins University, University of Edinburgh, and Imperial College London. He is the recipient of various awards and fellowships, notably, the Ferran Sunyer i Balaguer Prize in 2014 and 2018, Daiwa Adrian Prize in 2010, and the ISAAC award in 2007. He is serving as the head of the Ghent Analysis & PDE Center of Ghent University.

Berikbol Torebek is a Postdoctoral Researcher at the Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University in Belgium and Leading Researcher at the Institute of Mathematics and Mathematical Modelling in Kazakhstan. He is a member of the research group "Analysis and PDEs" led by Prof. Michael Ruzhansky. He obtained his PhD in Mathematics in July 2017 from the Al-Farabi Kazakh University. His research interests lie in classical and fractional Partial Differential Equations. In particular, he is interested in the solvability and qualitative properties of solutions to nonlinear problems. He is currently a member of the editorial boards of international journals "Fractional Differential Calculus", "International Journal of Mathematics and Physics" and "Kazakh Mathematical Journal".

**Karel Van Bockstal **obtained his PhD (in mathematical engineering) in 2015 at Ghent University, Belgium, and is currently a postdoctoral researcher (Ghent Analysis & PDE Center) at the Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University. His area of specialisation is related to mathematical analysis, evolutionary partial differential equations and the development of numerical algorithms and their implementation. This research focus concerns direct and inverse problems with applications in heat transfer, elasticity, electromagnetism and thermo-elasticity. He authors 37 publications included in ISI Web of Science. In addition, he was awarded the EAIP Young Scientist Award of the 8th International Conference "Inverse Problems: Modelling and Simulation", May 2016.

Inhalt

Part I Mini-courses.- The Hardy constant: a review.- Some harmonic analysis in a general Gaussian setting.- Introduction to hypercomplex analysis.- Some norm bounds for the spectral projections of the Heisenberg sublaplacian.- Boundary value problems for elliptic operators satisfying Carleson condition.- An elementary computation of heat trace invariants.- Can we divide vectors? - Geometric calculus in Science and Engineering.- Maximal regularity as a tool for partial differential equations.- Recent existence results for some critical subelliptic problems.- Introduction to the Analysis on Manifolds with Conical Singularities.- Elliptic systems of phase transition type.- Cordes condition, Campanato nearness and beyond.- Part II Short talks.- Global hypoellipticity on homogeneous vector bundles: Necessary and sufficient conditions.- On inverse source problems for space-dependent sources in thermoelasticity.- Symmetric properties of eigenvalues and eigenfunctions of uniform beams with axial loads.- Cylindrical and horizontal extensions of critical Sobolev type inequalities and identities.- Very weak solution of the wave equation for Sturm-Liouville operator.- Some new multidimensional Cochran-Lee and Hardy type inequalities.