We combine Spitzer and Herschel data of the star-forming region N11 in the Large Magellanic Cloud to produce detailed maps of the dust properties in the complex and study their variations with the ISM conditions. We also compare APEX/LABOCA 870um observations with our model predictions in order to decompose the 870um emission into dust and non-dust (free-free emission and CO(3-2) line) contributions. We find that in N11, the 870um can be fully accounted for by these 3 components. The dust surface density map of N11 is combined with HI and CO observations to study local variations in the gas-to-dust mass ratios. Our analysis leads to values lower than those expected from the LMC low-metallicity as well as to a decrease of the gas-to-dust mass ratio with the dust surface density. We explore potential hypotheses that could explain the low observed gas-to-dust mass ratios (variations in the XCO factor, presence of CO-dark gas or of optically thick HI or variations in the dust abundance in the dense regions). We finally decompose the local SEDs using a Principal Component Analysis (i.e. with no a priori assumption on the dust composition in the complex). Our results lead to a promising decomposition of the local SEDs in various dust components (hot, warm, cold) coherent with that expected for the region. Further analysis on a larger sample of galaxies will follow in order to understand how unique this decomposition is or how it evolves from one environment to another.