Doctor of Philosophy / Department of Mathematics / Ivan Blank / In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point [chi]₀ in the domain, there exists a nested family of sets { D[subscript]r([chi]₀) } where the average over any of those sets is related to the value of the function at [chi]₀. Although it is known that the { D[subscript]r([chi]₀) } are nested and are comparable to balls in the sense that there exists c, C depending only on L such that B[subscript]cr([chi]₀) ⊂ D[subscript]r([chi]₀) ⊂ B[subscript]Cr([chi]₀) for all r > 0 and [chi]₀ in the domain, otherwise their geometric and topological properties are largely unknown. In this work we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.

This thesis consists of an introduction and four research papers related to free boundary problems and systems of fully non-linear elliptic equations. Paper A and Paper B prove optimal regularity of solutions to general elliptic and parabolic free boundary problems, where the operators are fully non-linear and convex. Furthermore, it is proven that the free boundary is continuously differentiable around so called "thick" points, and that the free boundary touches the fixed boundary tangentially in two dimensions. Paper C analyzes singular points of solutions to perturbations of the unstable obstacle problem, in three dimensions. Blow-up limits are characterized and shown to be unique. The free boundary is proven to lie close to the zero-level set of the corresponding blow-up limit. Finally, the structure of the singular set is analyzed. Paper D discusses an idea on how existence and uniqueness theorems concerning quasi-monotone fully non-linear elliptic systems can be extended to systems that are not quasi-monotone. / <p>QC 20151210</p>

This paper considers a free energy functional and corresponding free boundary problem for multilayered structures which arise from a mixture of a block copolymer and a weak solvent. The free boundary problem is formally derived from the limit of large solvent/polymer segregation and intermediate segregation between monomer species. A change of variables based on Legendre transforms of the effective bulk energy is used to explicitly construct a family of equilibrium solutions. The second variation of the effective free energy of these solutions is shown to be positive. This result is used to show more generally that equilibria are local minimizers of the free energy.

We deal with a free boundary problem for a nonlinear parabolic equation, which includes a parameter in the free boundary condition. This type of system has been used in models of ecological systems, in chemical reactor theory and other kinds of propagation phenomena involving reactions and diffusion. The main purpose of this dissertation is to show the global existence, uniqueness of solutions and that a Hopf bifurcation occurs at a critical value of the parameter r. The existence and uniqueness of the solution for this problem are shown by finding an equivalent regular free boundary problem to which existence results can be applied. We then show that as the bifurcation parameter r decreases and passes through a critical value rc, the stationary solution loses stability and a stable periodic solution appears. Several figures have been included, which illustrate this transistion. The pascal source program used in the numerical simulation is included in an appendix.

No description available.

No description available.

We study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre. Specifically, we show that minimizers of the energy [mathematical equation] where [mathematical equations] with 0 < [gamma] < 1, with free behavior on the set {y=0}, are Holder continuous with exponent [Beta] = 2[sigma]/2-[gamma]. These minimizers exhibit a free boundary: along {y = 0}, they divide into a zero set {u = 0} and a positivity set where {u > 0}; we call the interface between these sets the free boundary. The regularity is optimal, due to the non-degeneracy property of the minimizers: in any ball of radius r centered at the free boundary, the minimizer grows (in the supremum sense) like r[Beta]. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. / text

We analyze the geometry and regularity of the largest subsolution of a Free Boundary Problem. We showed that the largest subsolution is a viscosity solution of (1) with Lipschitz and Non-Degenerate properties under a very general free boundary condition. In addition to this, we provide density bounds for the positivity set and its complement near the free boundary. / text

Implementation of the ultrasonic second harmonic generation has typically been restricted to simple setups such as through-transmission or Rayleigh surface waves. Recent research has evaluated the second harmonic generation in P- and SV- waves reflected from a stress-free surface to enable the single-sided interrogation of a specimen. This research considers the second harmonic generation in an aluminum specimen, which is analytically evaluated using an approach based on the perturbation method. Here, the model is chosen to mimic an experimental setup where a longitudinal wave is generated at an oblique angle and the reflected wave is detected using a set of wedge transducers. Due to mode conversion at the interface of the wedge and the specimen, it is necessary to evaluate longitudinal and shear waves, determining all second harmonic waves generated in the bulk and at the stressfree boundary. The theoretically developed model is then implemented in a commercial finite element code, COMSOL, using increasing fundamental wave amplitudes for different values of third order elastic constants. The results of this computational model verify the analytical approach and the proposed measurement setup, taking into account assumptions and approximations of the solution procedure. Furthermore, the computational model is used to draw important conclusions relevant to the experimental setup, including the need to avoid evolving surface waves and interactions with diffracted waves. These numerical results are used to develop a recommendation for the measurement position and incident angle. Finally, the nonlinearity of two different aluminum specimens is measured with the suggested measurement setup and the results confirm the feasibility of the single-sided determination of the acoustic nonlinearity using reflected bulk waves.

Doctor of Philosophy / Department of Mathematics / Ivan Blank / We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper ``The regularity of free boundaries in higher dimensions." Finally, we show that blowup limits are in general not unique at free boundary points.