Mathematics

The goal of the course is to cover the following topics:

• Basic calculus in Banach spaces
• Local inversion theorems and implicit function theory 
• Properties of bifurcation points 
• Bifurcation from the simple eigenvalue 
• Bifurcation from multiple eigenvalues
• Examples and applications 

Reference:

• Ambrosetti-Prodi, A primer in Nonlinear Analysis

Laplace equation:

• harmonic functions, mean value properties,
• maximum principle,
• Green's function,
• Poisson kernel,
• Harnack inequality,
• subharmonic functions,
• Perron-Wiener-Brelot method for the Dirichlet problem,
• regular boundary points.

Variational theory of elliptic equations:

1. Elements of convex analysis, polar and bipolar function and their properties, convex envelopes. Semiclassical theory, Euler-Lagrange equations and relation with elliptic PDE’s. Regularity of minimizers.
2. Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers.
3. Relaxation theorems, representation of relaxed functionals; convex, quasiconvex, polyconvex and rank-one convex envelopes.

1.  Optimal control problem. Existence of solution. Relaxation.
2.  Linear time-optimal problems.
3.  Linear-quadratic problems.
4.  Pontryagin Maximum Principle.
5.  Dubins car and Euler elasticae.
6.  Hamilton-Jacobi-Bellmann equation and dynamical programming.
7.  Chronological calculus.
8.  Lie groups and left-invariant problems.
9.  Sub-Riemannian problems on Lie groups and rolling balls.
10.  Second variation and Jacobi equation.
11.  Singular controls.
12.  Chattering controls.

• The cell and its parts
• Mechanics of the plasma membrane
• Mechanics of the cytoskeleton
• Mechanics of adhesion
• Mechanotransduction