Schr\"odinger-Poisson equations with singular potentials in R3R^3

The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schr\"{o}dinger-Poisson system {ll} -\Delta u +(\lambda+\frac{1}{|y|^\alpha})u+\phi (x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -\Delta\phi = u^2,\ \lim\limits_{|x|\rightarrow +\infty}\phi(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $\lambda\geqslant0$, $\alpha\in[0,8)$ and $\max\{2,\frac{2+\alpha}{2}\}<p<5$.Comment: 23page

Engineering 3D architected metamaterials for enhanced mechanical properties and functionalities.

Compared with conventional materials, architected metamaterials have shown unprecedented mechanical properties and functionalities applications. Featured with controlled introduction of porosity and different composition, architected metamaterials have demonstrated unprecedent properties not found in natural materials. Such design strategies enable researchers to tailor materials and structures with multifunctionalies and satisfy conflicting design requirements, such as high stiffness and toughness; high strength with vibration mitigation properties, etc. Furthermore, with the booming advancement of 3D printing technologies, architected materials with precisely defined complex topologies can be fabricated effortlessly, which in turn promotes the research significantly. The research objectives of this dissertation are to achieve the enhanced mechanical properties and multifunctionalities of architected metamaterials by integrative design, computational modeling, 3D printing, and mechanical testing. Phononic crystal materials are capable of prohibiting the propagation of mechanical waves in certain frequency ranges. This certain frequency ranges are represented by phononic band gaps. Formally, band gaps are formed through two main mechanisms, Bragg scattering and local resonance. Band gaps induced by Bragg scattering are dependent on periodicity and the symmetry of the lattice. However, phononic crystals with Bragg-type band gaps are limited in their application because they do not attenuate vibration at lower frequencies without requiring large geometries. It is not practical to build huge models to achieve low frequency vibration mitigation. Alternatively, band gaps formed by local resonance are due to the excitation of resonant frequencies, and these band gaps are independent of periodicity. Therefore, lower frequency band gaps have been explored mostly through the production of phononic metamaterials that exploit locally resonant masses to absorb vibrational energy. However, despite research advances, the application of phononic metamaterials is sill largely hindered by their limited operation frequency ranges. Designing lightweight phononic metamaterials with low-frequency vibration mitigation capability is still a challenging topic. On the other hand, conventional phononic crystals usually exhibit very poor mechanical properties, such as low stiffness, strength, and energy absorption. This could largely limit their practical applications. Ideally, multifunctional materials and structures with both vibration mitigation property and high mechanical performance are demanded. In this work, we propose architected polymer foam material to overcome the challenges. Beside altering the topological architecture of metamaterials, tailoring the composition of materials is another approach to enhance the mechanical properties and realize multifunctionalities. Natural materials have adopted this strategy for long period of time. Biological structural materials such as nacre, glass sea sponges feature unusual mechanical properties due to the synergistic interplay between hard and soft material phases. These exceptional mechanical performance are highly demanded in engineering applications. As such, intensive efforts have been devoted to developing lightweight structural composites to meet the requirements. Despite the significant advances in research, the design and fabrication of low-cost structural materials with lightweight and superior mechanical performance still represent a challenge. Taking inspiration from cork material, we propose a new type of multilayered cellular composite (MCC) structure composed of hard brittle and soft flexible phases to tackle this challenge. On the other hand, piezoelectric materials with high sensitivity but low energy absorption have largely limited their applications, especially during harsh environment where external load could significantly damage the materials. Enlightened by the multiphase composite concept, we apply this design motif to develop a new interpenetrating-phased piezoelectric materials by combining PZT material as skeleton and PDMS material as matrix. By using a facial camphene-templated freeze-casting method, the co-continuous composites are fabricated with good quality. Through experiment and simulation studies, the proposed composite demonstrates multifunction with exceptional energy absorption and high sensitivity. Based on the above experimental studies, we further propose to use topology optimization framework to obtain the composites with the best performance of multifunctionalities. Specifically, we will use the solid isotropic material with penalization (SIMP) approach to optimize the piezoelectric materials with multi-objectives of 1) energy absorption and 2) electric-mechanical conversion property. The materials for the optimization design will be elastic PZT as skeleton and elatic material PDMS as matrix. To enable the gradient search of objective function efficiently, we will use adjoint method to derive the shape sensitivity analysis