Energy solutions to one-dimensional singular parabolic problems with BVBV data are viscosity solutions

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in $BV$, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.Comment: 15 page

Very Singular Diffusion Equations-Second and Fourth Order Problems

This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example

Understanding non-compliance to colorectal cancer screening: a case control study, nested in a randomised trial [ISRCTN83029072]

BACKGROUND: The major limit to colorectal cancer screening effectiveness is often low compliance. We studied the reasons for non compliance and determinants of compliance to faecal occult blood tests in Lazio, Italy. METHODS: This is a case-control study nested within a trial that tested the effect of type of test and provider on colorectal cancer screening compliance. Non compliant trial subjects were classified as cases, and compliant subjects were classified as controls. We sampled 600 cases and 600 controls matched by their general practitioner, half were invited for screening at the hospital, and the other half directly at their general practitioner's office. Cases and controls answered questions on: distance from test provider, logistical problems, perception of colorectal cancer risk, confidence in screening efficacy, fear of results, presence of colorectal cancer in the family, and gastrointestinal symptoms. RESULTS: About 31% of cases never received the letter offering free screening, and 17% of the sampled population had already been screened. The first reported reason for non-compliance was "lack of time" (30%); the major determinant of compliance was the distance from the test provider: odds ratio >30 minutes vs <15 minutes 0.3 (95%CI = 0.2–0.7). The odds ratio for lack of time was 0.16 (95% IC 0.1–0.26). The effect was stronger if the hospital (0.03 95%CI = 0.01–0.1) rather than the general practitioner (0.3 95%CI = 0.2–0.6) was the provider. Twenty-two percent of controls were accompanied by someone to the test. CONCLUSION: To increase compliance, screening programmes must involve test providers who are geographically close to the target population

A variational view at the time-dependent minimal surface equation

We present a global variational approach to the L2-gradient flow of the area functional of cartesian surfaces through the study of the so-called weighted energy-dissipation (WED) functional. In particular, we prove a relaxation result which allows us to show that minimizers of the WED converge in a quantitatively prescribed way to gradient-flow trajectories of the relaxed area functional. The result is then extended to general parabolic quasilinear equations arising as gradient flows of convex functionals with linear growth