Dirichlet problem for Poisson equations in Jordan domains

First, we study the Dirichlet problem for the Poisson equations \(\triangle u(z) = g(z)\) with \(g\in L^p\), \(p>1\), and continuous boundary data \(\varphi :\partial D\to\mathbb{R}\) in arbitrary Jordan domains \(D\) in \(\mathbb{C}\) and prove the existence of continuous solutions \(u\) of the problem in the class \(W^{2,p}_{\rm loc}\). Moreover, \(u\in W^{1,q}_{\rm loc}\) for some \(q>2\) and \(u\) is locally Hölder continuous. Furthermore, \(u\in C^{1,\alpha}_{\rm loc}\) with \(\alpha = (p-2)/p\) if \(p>2\). Then, on this basis and applying the Leray-Schauder approach, we obtain the similar results for the Dirichlet problem with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form \(\triangle u(z) = h(z)\cdot f(u(z))\) with the same assumptions on \(h\) as for \(g\) above and continuous functions \(f:\mathbb{R}\to\mathbb{R}\), either bounded or with nondecreasing \(|f\,|\) of \( |t\,|\) such that \(f(t)/t \to 0\) as \(t\to\infty\). We also give here applications to mathematical physics that are relevant to problems of diffusion with absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson equations in the unit disk \(\mathbb{D}\subset\mathbb{C}\) with arbitrary boundary data \(\varphi :\partial\mathbb{D}\to\mathbb{R}\) that are measurable with respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions \(u\) of the problem in terms of the angular limits in \(\mathbb D\) a.e. on \(\partial\mathbb D\) with respect to logarithmic capacity with the same local properties as above. Finally, we extend these results to almost smooth Jordan domains with qusihyperbolic boundary condition by Gehring-Martio.

FUNCTIONAL ELBOW AND WRIST REHABILITATION PROTOTYPE CONTROLLED BY COMPUTER

On this project, we have considered the anthropometrics of the human arm, taking in consideration the movements and angles of the elbow and wrist by developing a prototype for the first and second phases of rehabilitation as the main goal the patient could achieve full joints mobility. The implementation of this prototype consists in four different adaptations, one for each movement, an interface electronic board of sensors, a control board, and a graphical interface of the user where the physiotherapist is able to set up a personalized rehabilitation cycle responding to the patient needs. We have done field tests of the prototype with a patient with elbow and wrist fracture diagnosis. The results of these tests showed an improvement in the mobility of both joints through a small number of rehabilitation sessions. Thus, it is concluded that the prototype allows the patient the progressive reach of angles nearer to the angular limits of pronation-supination of the elbow and flexion-extension of the wrist, with a reduction of 50% of number of session using conventional methods.

On Angular Limits of Normal Meromorphic Functions: A Geometric Aspect

We will prove the assertions which give necessary and sufficient conditions for a normal meromorphic function on the open unit disk to have an angular limit. The results obtained show that the conditions from the classical Lindelöf theorem, as well as the theorems of Lehto and Virtanen and Bagemihl and Seidel, concerning angular limit values of meromorphic functions, can be weakened.

Shaping Patient Specific Surgical Guides for Arthroplasty to Obtain High Docking Robustness

Patient specific surgical guides (PSSGs) are used in joint replacement surgery to simplify the surgical process and to increase the accuracy in alignment of implant components with respect to the bone. Each PSSG is fabricated patient specifically and fits only in the planned position on the joint surface by the matching shape. During surgery, the surgeon holds the PSSG in the planned position and the incorporated guidance is used in making the essential cuts to fit the implant components. The shape of the PSSG determines its docking robustness (i.e., the range of forces that the surgeon may apply without losing the planned position). Minimal contact between the PSSG and the joint surface is desired, as this decreases the likelihood of interposition with undetected tissues. No analytical method is known from literature where the PSSG shape can be optimized to have high docking robustness and minimal bone-guide contact. Our objective is to develop and validate such an analytical method. The methods of motion restraint, moment labeling and wrench space—applied in robotic grasping and workpart fixturing—are employed in the creation of this new method. The theoretic approach is utilized in an example by optimizing the PSSG shape for one joint surface step-by-step. The PSSGs that arise from these optimization steps are validated with physical experiments. The following design tools for the analytical method are introduced. The optimal location for bone-guide contact and the application surface where the surgeon may push can be found graphically, respectively, by the use of the wrench space map and the application angle map. A quantitative analysis can be conducted using the complementary wrench space metrics and the robustness metric R. Utilization of the analytical method with an example joint surface shows that the PSSG's shape can be optimized. Experimental validation shows that the standard deviation of the error between the measured and calculated angular limits in the docking force is only 0.7 deg. The analytical method provides valid results and thus can be used for the design of PSSGs.