On the geometry of the pp-Laplacian operator

The $p$-Laplacian operator $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet and eigenvalue problems associated with the $p$-Laplacian lead to intriguing geometric questions, because their limits for $p\to\infty$ or $p\to 1$ can be characterized by the geometry of $\Omega$. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general $p\in[1,\infty]$. We report also on results concerning the normalized or game-theoretic $p$-Laplacian $$\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$$ and its parabolic counterpart $u_t-\Delta_p^N u=0$. These equations are homogeneous of degree 1 and $\Delta_p^N$ is uniformly elliptic for any $p\in (1,\infty)$. In this respect it is more benign than the $p$-Laplacian, but it is not of divergence type.Comment: 15 pages, 5 figures, Survey lecture given at the WIAS conference "Theory and Applications of Partial Differential Equations" in Dec. 201

A study on gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations

We investigate sharp conditions for boundary and interior gradient es- timates of continuous viscosity solutions to fully nonlinear, uniformly ellip- tic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain

Влияние полиненасыщенных жирных кислот на биохимическую активность комбинированной закваски

В данной работе представлены результаты культивирования комбинированнойзакваски, соотношение культур Lactobacillus helveticus 35-1 и Propionibakterium freudenreichii subsp. shermanii АС 2503 в которой 40:60, на обезжиренном молоке, с добавлением рыжикового масла. Подобрана оптимальная доза внесения рыжикового масла, обеспечивающая более высокую кислотообразующую способность и количество жизнеспособных клеток в комбинированной закваске

Effects of clinical decision topic on patients’ involvement in and satisfaction with decisions and their subsequent implementation

Clinical decision-making is the vehicle for mental health care delivery, and predictors of decision-making experience and adherence are under-researched. The aim was to investigate the relationship between decision topic and kind of involvement in the decision, satisfaction and subsequent implementation, from both staff and patient perspectives

Clinical decision making and mental health service use in people with severe mental illness across Europe

Objective: This study aims to explore relationships between preferred and experienced clinical decision making with service use, and associated costs, by people with severe mental illness. Methods: Prospective observational study of mental healthcare in six European countries: Germany, UK, Italy Hungary, Denmark and Switzerland. Patients (N = 588) and treating clinicians (N = 213) reported preferred and experienced decision making at baseline using the Clinical Decision Making Style Scale (CDMS) and the Clinical Decision Involvement and Satisfaction Scale (CDIS). Retrospective service use was assessed with the Client Socio-Demographic and Service Receipt Inventory (CSSRI-EU) at baseline and 12-month follow-up. Negative binomial regression analyses examined the effects of CDMS and CDIS on service use and inpatient costs at baseline and multilevel models examined these relationships over time. Results: At baseline, staff and patient preferences for active decision making and low patient satisfaction with experienced decisions were associated with longer hospital admissions and higher costs. Patient preferences for active decision making predicted increases in hospital admissions (b = .236, p =.043) over 12 months and cost increases were predicted by low patient satisfaction (b = 4803, p =.005). Decision making was unrelated to medication, outpatient, or community service use. Conclusions: Decision making is related to inpatient service use and associated costs by people with severe mental illness. A preference for shared decision making may reduce healthcare costs via a reduction in inpatient admissions. Patient satisfaction with decisions is a crucial predictor of healthcare costs; therefore, clinicians should maximize patient satisfaction with decision making

Helping alliance and unmet needs in routine care of people with severe mental illness across Europe: a prospective longitudinal multicenter study

The helping alliance (HA) refers to the collaborative bond between patient and therapist including shared goals and tasks. People with severe mental illness have a complex mixture of clinical and social needs. Using mixed-effects regression, this study examined in 588 people with severe mental illness whether an increase in the HA is associated with fewer unmet needs over time, and whether change in the HA precedes change in unmet needs. It was found that a reduction of unmet needs was slower in patients with higher HA (B=0.04, p<.0001) only for patient-rated measures. Improvement in both patient-rated and staff-rated HA over time was associated with fewer subsequent patient- (B=-0.10, p<0.0001) and staff-rated (B=-0.08, p=0.0175) unmet needs. With positive changes in the HA preceding fewer unmet needs, findings provide further evidence for a causal relationship between alliance and outcome in the treatment of people with severe mental illness

Clinical Decision Making and Outcome in Routine Care for People with Severe Mental Illness (CEDAR): Study protocol

BACKGROUND: A considerable amount of research has been conducted on clinical decision making (CDM) in short-term physical conditions. However, there is a lack of knowledge on CDM and its outcome in long-term illnesses, especially in care for people with severe mental illness. METHODS/DESIGN: The study entitled "Clinical decision making and outcome in routine care for people with severe mental illness" (CEDAR) is carried out in six European countries (Denmark, Germany, Hungary, Italy, Switzerland and UK). First, CEDAR establishes a methodology to assess CDM in people with severe mental illness. Specific instruments are developed (and psychometric properties established) to measure CDM style, key elements of CDM in routine care, as well as CDM involvement and satisfaction from patient and therapist perspectives. Second, these instruments are being put to use in a multi-national prospective observational study (bimonthly assessments during a one-year observation period; N = 560). This study investigates the immediate, short- and long-term effect of CDM on crucial dimensions of clinical outcome (symptom level, quality of life, needs) by taking into account significant variables moderating the relationship between CDM and outcome. DISCUSSION: The results of this study will make possible to delineate quality indicators of CDM, as well as to specify prime areas for further improvement. Ingredients of best practice in CDM in the routine care for people with severe mental illness will be extracted and recommendations formulated. With its explicit focus on the patient role in CDM, CEDAR will also contribute to strengthening the service user perspective. This project will substantially add to improving the practice of CDM in mental health care across Europe. TRIAL REGISTER: ISRCTN75841675