Narrow Results

Certain financial investments have different profitabilities according to the invested capital. In particular, there are some bank transactions, such as progressive current accounts, which discriminate nominal rates of interest, depending on the invested sums, that is, transactions whose underlying financial laws are not homogeneous of the first degree with respect to the amounts. More specifically, this discrimination occurs when assigning an equal rate to the capitals C in the same interval ]C_i,C_i+1]. This makes the financial law discontinuous with finite jumps, once the term has been fixed. Of course, it would be convenient, for a group of investors, to join their savings because greater rates of interest can be obtained. The question is how to distribute, in a rational way or with equity, among the individual agents, the interest obtained jointly. Our findings are based on a progressive sharing, using differential calculus.

In this paper, we study an evolution equation involving the normalized $p$-Laplacian and a bounded continuous source term. The normalized $p$-Laplacian is in non-divergence form and arises for example from stochastic tug-of-war games with noise. We prove local $C^{\alpha ,\frac{\alpha }{2}}$ regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.

Models for shallow water wave processes are routinely applied in coastal, estuarine and river engineering practice, to problems such as flood waves, tidal circulation, tsunami penetration, and storm tides. Evaluation and confirmation of application codes can be facilitated by analytical solutions that may represent this wide range of physical problems, without significantly compromising the contributing flow processes. The appropriate linearized shallow water wave equations are defined, together with quite general initial and boundary conditions that will represent the expected variety of shallow water wave problems. A completely general analytical solution for both water surface elevation and flow is established, in a manner suitable for the evaluation of shallow water wave codes. The contribution of both free and forced modes is identified, together with the role of friction.