Modern Problems in PDEs and Applications

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.

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.